%------------------------------------------------------------------------------
% File     : ITP282_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_BuildupMemImp 00337_015688
% 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    : 0093_VEBT_BuildupMemImp_00337_015688 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11532 (4400 unt;1333 typ;   0 def)
%            Number of atoms       : 18893 (8648 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 19320 (2085   ~; 316   |;2050   &)
%                                         (2052 <=>;12817  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 1576 ( 679 fml; 897 var)
%            Number of X terms     : 1031 (   0  []; 763 ite; 268 let)
%            Number of types       :   23 (  22 usr)
%            Number of type conns  : 1161 ( 955   >; 206   *;   0   +;   0  <<)
%            Number of predicates  :  226 ( 223 usr;   2 prp; 0-13 aty)
%            Number of functors    : 1114 (1114 usr;  90 con; 0-8 aty)
%            Number of variables   : 30574 (27787   !; 885   ?;30574   :)
%                                         (1902  !>;   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 18:08:42.847
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
tff(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
    vEBT_VEBTi: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_t_String_Oliteral,type,
    literal: $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_Sum__Type_Osum,type,
    sum_sum: ( $tType * $tType ) > $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 (1295)
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_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_Groups_Ominus,type,
    minus: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Cardinality_Ocard2,type,
    card2: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

tff(sy_cl_Cardinality_OCARD__1,type,
    cARD_1: 
      !>[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_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_Limits_Otopological__group__add,type,
    topolo1633459387980952147up_add: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ouniform__space,type,
    topolo7287701948861334536_space: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type,
    real_V768167426530841204y_dist: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__bh____,type,
    aTP_Lamp_bh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bi____,type,
    aTP_Lamp_bi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bj____,type,
    aTP_Lamp_bj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bk____,type,
    aTP_Lamp_bk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bl____,type,
    aTP_Lamp_bl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bm____,type,
    aTP_Lamp_bm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__bp____,type,
    aTP_Lamp_bp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bq____,type,
    aTP_Lamp_bq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__br____,type,
    aTP_Lamp_br: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bs____,type,
    aTP_Lamp_bs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bt____,type,
    aTP_Lamp_bt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bu____,type,
    aTP_Lamp_bu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bv____,type,
    aTP_Lamp_bv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bw____,type,
    aTP_Lamp_bw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bx____,type,
    aTP_Lamp_bx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__by____,type,
    aTP_Lamp_by: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bz____,type,
    aTP_Lamp_bz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ca____,type,
    aTP_Lamp_ca: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cb____,type,
    aTP_Lamp_cb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cc____,type,
    aTP_Lamp_cc: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ci____,type,
    aTP_Lamp_ci: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__cj____,type,
    aTP_Lamp_cj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ck____,type,
    aTP_Lamp_ck: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cl____,type,
    aTP_Lamp_cl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cm____,type,
    aTP_Lamp_cm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cn____,type,
    aTP_Lamp_cn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__co____,type,
    aTP_Lamp_co: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cp____,type,
    aTP_Lamp_cp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

tff(sy_c_ATP_058Lamp__de____,type,
    aTP_Lamp_de: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__df____,type,
    aTP_Lamp_df: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dg____,type,
    aTP_Lamp_dg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dh____,type,
    aTP_Lamp_dh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__di____,type,
    aTP_Lamp_di: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dj____,type,
    aTP_Lamp_dj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dk____,type,
    aTP_Lamp_dk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dl____,type,
    aTP_Lamp_dl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dm____,type,
    aTP_Lamp_dm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dn____,type,
    aTP_Lamp_dn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__do____,type,
    aTP_Lamp_do: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__eh____,type,
    aTP_Lamp_eh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ei____,type,
    aTP_Lamp_ei: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ej____,type,
    aTP_Lamp_ej: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ek____,type,
    aTP_Lamp_ek: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__el____,type,
    aTP_Lamp_el: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__em____,type,
    aTP_Lamp_em: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__en____,type,
    aTP_Lamp_en: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eo____,type,
    aTP_Lamp_eo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ep____,type,
    aTP_Lamp_ep: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eq____,type,
    aTP_Lamp_eq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__er____,type,
    aTP_Lamp_er: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__es____,type,
    aTP_Lamp_es: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__et____,type,
    aTP_Lamp_et: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eu____,type,
    aTP_Lamp_eu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ev____,type,
    aTP_Lamp_ev: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ew____,type,
    aTP_Lamp_ew: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ex____,type,
    aTP_Lamp_ex: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ey____,type,
    aTP_Lamp_ey: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ez____,type,
    aTP_Lamp_ez: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fa____,type,
    aTP_Lamp_fa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fb____,type,
    aTP_Lamp_fb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fc____,type,
    aTP_Lamp_fc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fd____,type,
    aTP_Lamp_fd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fe____,type,
    aTP_Lamp_fe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ff____,type,
    aTP_Lamp_ff: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fg____,type,
    aTP_Lamp_fg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fh____,type,
    aTP_Lamp_fh: 
      !>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(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] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__hb____,type,
    aTP_Lamp_hb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hc____,type,
    aTP_Lamp_hc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hd____,type,
    aTP_Lamp_hd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__he____,type,
    aTP_Lamp_he: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hf____,type,
    aTP_Lamp_hf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hg____,type,
    aTP_Lamp_hg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hh____,type,
    aTP_Lamp_hh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hi____,type,
    aTP_Lamp_hi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hj____,type,
    aTP_Lamp_hj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hk____,type,
    aTP_Lamp_hk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hl____,type,
    aTP_Lamp_hl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hm____,type,
    aTP_Lamp_hm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hn____,type,
    aTP_Lamp_hn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ho____,type,
    aTP_Lamp_ho: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hp____,type,
    aTP_Lamp_hp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hq____,type,
    aTP_Lamp_hq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hr____,type,
    aTP_Lamp_hr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hs____,type,
    aTP_Lamp_hs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ht____,type,
    aTP_Lamp_ht: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hu____,type,
    aTP_Lamp_hu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hv____,type,
    aTP_Lamp_hv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hw____,type,
    aTP_Lamp_hw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hx____,type,
    aTP_Lamp_hx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hy____,type,
    aTP_Lamp_hy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hz____,type,
    aTP_Lamp_hz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ia____,type,
    aTP_Lamp_ia: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ib____,type,
    aTP_Lamp_ib: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ic____,type,
    aTP_Lamp_ic: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__id____,type,
    aTP_Lamp_id: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ie____,type,
    aTP_Lamp_ie: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__if____,type,
    aTP_Lamp_if: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ig____,type,
    aTP_Lamp_ig: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ih____,type,
    aTP_Lamp_ih: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ii____,type,
    aTP_Lamp_ii: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ij____,type,
    aTP_Lamp_ij: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ik____,type,
    aTP_Lamp_ik: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__il____,type,
    aTP_Lamp_il: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__im____,type,
    aTP_Lamp_im: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__in____,type,
    aTP_Lamp_in: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__io____,type,
    aTP_Lamp_io: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ip____,type,
    aTP_Lamp_ip: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iq____,type,
    aTP_Lamp_iq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ir____,type,
    aTP_Lamp_ir: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__is____,type,
    aTP_Lamp_is: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__it____,type,
    aTP_Lamp_it: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iu____,type,
    aTP_Lamp_iu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iv____,type,
    aTP_Lamp_iv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iw____,type,
    aTP_Lamp_iw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ix____,type,
    aTP_Lamp_ix: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iy____,type,
    aTP_Lamp_iy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iz____,type,
    aTP_Lamp_iz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ja____,type,
    aTP_Lamp_ja: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jb____,type,
    aTP_Lamp_jb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jc____,type,
    aTP_Lamp_jc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jd____,type,
    aTP_Lamp_jd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__je____,type,
    aTP_Lamp_je: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jf____,type,
    aTP_Lamp_jf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jg____,type,
    aTP_Lamp_jg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jh____,type,
    aTP_Lamp_jh: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ju____,type,
    aTP_Lamp_ju: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jv____,type,
    aTP_Lamp_jv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

tff(sy_c_ATP_058Lamp__kp____,type,
    aTP_Lamp_kp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kq____,type,
    aTP_Lamp_kq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__kr____,type,
    aTP_Lamp_kr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ks____,type,
    aTP_Lamp_ks: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kt____,type,
    aTP_Lamp_kt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ku____,type,
    aTP_Lamp_ku: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ly____,type,
    aTP_Lamp_ly: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lz____,type,
    aTP_Lamp_lz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ma____,type,
    aTP_Lamp_ma: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__mi____,type,
    aTP_Lamp_mi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mj____,type,
    aTP_Lamp_mj: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ol____,type,
    aTP_Lamp_ol: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__om____,type,
    aTP_Lamp_om: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__on____,type,
    aTP_Lamp_on: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oo____,type,
    aTP_Lamp_oo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__op____,type,
    aTP_Lamp_op: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oq____,type,
    aTP_Lamp_oq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__or____,type,
    aTP_Lamp_or: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__os____,type,
    aTP_Lamp_os: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ot____,type,
    aTP_Lamp_ot: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ou____,type,
    aTP_Lamp_ou: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ov____,type,
    aTP_Lamp_ov: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ow____,type,
    aTP_Lamp_ow: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ox____,type,
    aTP_Lamp_ox: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oy____,type,
    aTP_Lamp_oy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__pa____,type,
    aTP_Lamp_pa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pb____,type,
    aTP_Lamp_pb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pc____,type,
    aTP_Lamp_pc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pd____,type,
    aTP_Lamp_pd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pe____,type,
    aTP_Lamp_pe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pf____,type,
    aTP_Lamp_pf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pg____,type,
    aTP_Lamp_pg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ph____,type,
    aTP_Lamp_ph: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pi____,type,
    aTP_Lamp_pi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pj____,type,
    aTP_Lamp_pj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pk____,type,
    aTP_Lamp_pk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pl____,type,
    aTP_Lamp_pl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pm____,type,
    aTP_Lamp_pm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pn____,type,
    aTP_Lamp_pn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__po____,type,
    aTP_Lamp_po: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pp____,type,
    aTP_Lamp_pp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pq____,type,
    aTP_Lamp_pq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pr____,type,
    aTP_Lamp_pr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ps____,type,
    aTP_Lamp_ps: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pt____,type,
    aTP_Lamp_pt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pu____,type,
    aTP_Lamp_pu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pv____,type,
    aTP_Lamp_pv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pw____,type,
    aTP_Lamp_pw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__px____,type,
    aTP_Lamp_px: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__py____,type,
    aTP_Lamp_py: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pz____,type,
    aTP_Lamp_pz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qa____,type,
    aTP_Lamp_qa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qb____,type,
    aTP_Lamp_qb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qc____,type,
    aTP_Lamp_qc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qd____,type,
    aTP_Lamp_qd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qe____,type,
    aTP_Lamp_qe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qf____,type,
    aTP_Lamp_qf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qg____,type,
    aTP_Lamp_qg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qh____,type,
    aTP_Lamp_qh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qi____,type,
    aTP_Lamp_qi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qj____,type,
    aTP_Lamp_qj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qk____,type,
    aTP_Lamp_qk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ql____,type,
    aTP_Lamp_ql: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qm____,type,
    aTP_Lamp_qm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qn____,type,
    aTP_Lamp_qn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qo____,type,
    aTP_Lamp_qo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qp____,type,
    aTP_Lamp_qp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qq____,type,
    aTP_Lamp_qq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qr____,type,
    aTP_Lamp_qr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qs____,type,
    aTP_Lamp_qs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qt____,type,
    aTP_Lamp_qt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qu____,type,
    aTP_Lamp_qu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qv____,type,
    aTP_Lamp_qv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qw____,type,
    aTP_Lamp_qw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qx____,type,
    aTP_Lamp_qx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qy____,type,
    aTP_Lamp_qy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qz____,type,
    aTP_Lamp_qz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ra____,type,
    aTP_Lamp_ra: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rb____,type,
    aTP_Lamp_rb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rc____,type,
    aTP_Lamp_rc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rd____,type,
    aTP_Lamp_rd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__re____,type,
    aTP_Lamp_re: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rf____,type,
    aTP_Lamp_rf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rg____,type,
    aTP_Lamp_rg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rh____,type,
    aTP_Lamp_rh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ri____,type,
    aTP_Lamp_ri: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rj____,type,
    aTP_Lamp_rj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rk____,type,
    aTP_Lamp_rk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rl____,type,
    aTP_Lamp_rl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rm____,type,
    aTP_Lamp_rm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ro____,type,
    aTP_Lamp_ro: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rp____,type,
    aTP_Lamp_rp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rq____,type,
    aTP_Lamp_rq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rr____,type,
    aTP_Lamp_rr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__rt____,type,
    aTP_Lamp_rt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ru____,type,
    aTP_Lamp_ru: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rv____,type,
    aTP_Lamp_rv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rw____,type,
    aTP_Lamp_rw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rx____,type,
    aTP_Lamp_rx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ry____,type,
    aTP_Lamp_ry: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rz____,type,
    aTP_Lamp_rz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sa____,type,
    aTP_Lamp_sa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sb____,type,
    aTP_Lamp_sb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sc____,type,
    aTP_Lamp_sc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sd____,type,
    aTP_Lamp_sd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__sf____,type,
    aTP_Lamp_sf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sg____,type,
    aTP_Lamp_sg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sh____,type,
    aTP_Lamp_sh: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tg____,type,
    aTP_Lamp_tg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__th____,type,
    aTP_Lamp_th: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ti____,type,
    aTP_Lamp_ti: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tj____,type,
    aTP_Lamp_tj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tk____,type,
    aTP_Lamp_tk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tl____,type,
    aTP_Lamp_tl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tm____,type,
    aTP_Lamp_tm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tn____,type,
    aTP_Lamp_tn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__to____,type,
    aTP_Lamp_to: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tp____,type,
    aTP_Lamp_tp: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vm____,type,
    aTP_Lamp_vm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vn____,type,
    aTP_Lamp_vn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vo____,type,
    aTP_Lamp_vo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vp____,type,
    aTP_Lamp_vp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vq____,type,
    aTP_Lamp_vq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vr____,type,
    aTP_Lamp_vr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vs____,type,
    aTP_Lamp_vs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vt____,type,
    aTP_Lamp_vt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vu____,type,
    aTP_Lamp_vu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vv____,type,
    aTP_Lamp_vv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vw____,type,
    aTP_Lamp_vw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vx____,type,
    aTP_Lamp_vx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vy____,type,
    aTP_Lamp_vy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vz____,type,
    aTP_Lamp_vz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wa____,type,
    aTP_Lamp_wa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wb____,type,
    aTP_Lamp_wb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wc____,type,
    aTP_Lamp_wc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wd____,type,
    aTP_Lamp_wd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__we____,type,
    aTP_Lamp_we: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wf____,type,
    aTP_Lamp_wf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wg____,type,
    aTP_Lamp_wg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wh____,type,
    aTP_Lamp_wh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wi____,type,
    aTP_Lamp_wi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wj____,type,
    aTP_Lamp_wj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wk____,type,
    aTP_Lamp_wk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wl____,type,
    aTP_Lamp_wl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wm____,type,
    aTP_Lamp_wm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wn____,type,
    aTP_Lamp_wn: 
      !>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xp____,type,
    aTP_Lamp_xp: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : fun(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_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_Omake,type,
    array_make: 
      !>[A: $tType] : ( ( nat * fun(nat,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) * nat ) > 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_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 > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).

tff(sy_c_Assertions_Oentails,type,
    entails: ( assn * 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__QUERY,type,
    fI_QUERY: ( assn * assn * assn ) > $o ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits,type,
    bit_bi4170147762399595738t_bits: 
      !>[A: $tType] : ( fun(nat,$o) > A ) ).

tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: fun(nat,$o) > $o ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int ) > fun(int,int) ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : ( ( nat * 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] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,$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_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_Odup,type,
    code_dup: code_integer > code_integer ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun(num,int) ).

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_Complex_OArg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: complex > complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: complex > real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: complex > real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Deriv_Odifferentiable,type,
    differentiable: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__derivative,type,
    has_derivative: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__field__derivative,type,
    has_field_derivative: 
      !>[A: $tType] : ( ( fun(A,A) * A * filter(A) ) > $o ) ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : ( set(B) > nat ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),$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_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] : ( ( A * nat * $o ) > A ) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : ( A > fun(A,A) ) ).

tff(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

tff(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : ( 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_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

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_Oreturn,type,
    heap_Time_return: 
      !>[A: $tType] : ( A > heap_Time_Heap(A) ) ).

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_Int_Onat,type,
    nat2: int > nat ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Least__significant__bit_Olsb__class_Olsb,type,
    least_8051144512741203767sb_lsb: 
      !>[A: $tType] : fun(A,$o) ).

tff(sy_c_Limits_OBfun,type,
    bfun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : fun(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_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

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) > fun(list(A),list(A)) ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * 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] : ( 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) > fun(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_Olistset,type,
    listset: 
      !>[A: $tType] : ( list(set(A)) > set(list(A)) ) ).

tff(sy_c_List_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).

tff(sy_c_List_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

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

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oproduct__lists,type,
    product_lists: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

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

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Misc_OEps__Opt,type,
    eps_Opt: 
      !>[A: $tType] : ( fun(A,$o) > option(A) ) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),$o) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_Oslice,type,
    slice: 
      !>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
    most_s684356279273892711sb_msb: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Multiset_Ois__empty,type,
    is_empty: 
      !>[A: $tType] : ( multiset(A) > $o ) ).

tff(sy_c_Multiset_Olinorder__class_Opart,type,
    linorder_part: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

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

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_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_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Numeral__Type_OAbs__bit0_H,type,
    numeral_Abs_bit0: 
      !>[A: $tType] : ( int > numeral_bit0(A) ) ).

tff(sy_c_Numeral__Type_OAbs__bit1_H,type,
    numeral_Abs_bit1: 
      !>[A: $tType] : ( int > numeral_bit1(A) ) ).

tff(sy_c_Numeral__Type_Obit0_OAbs__bit0,type,
    numeral_Abs_bit02: 
      !>[A: $tType] : fun(int,numeral_bit0(A)) ).

tff(sy_c_Numeral__Type_Obit0_ORep__bit0,type,
    numeral_Rep_bit0: 
      !>[A: $tType] : ( numeral_bit0(A) > int ) ).

tff(sy_c_Numeral__Type_Obit1_OAbs__bit1,type,
    numeral_Abs_bit12: 
      !>[A: $tType] : fun(int,numeral_bit1(A)) ).

tff(sy_c_Numeral__Type_Obit1_ORep__bit1,type,
    numeral_Rep_bit1: 
      !>[A: $tType] : fun(numeral_bit1(A),int) ).

tff(sy_c_Numeral__Type_Omod__type,type,
    numeral_mod_type: 
      !>[A: $tType] : ( ( int * fun(A,int) * fun(int,A) ) > $o ) ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),A) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : ( A > fun(A,A) ) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

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] : ( A > fun(B,product_prod(A,B)) ) ).

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_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_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_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : ( real > A ) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > fun(A,A) ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : ( ( A * 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] : ( ( A * 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) * A ) > $o ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( fun(A,$o) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > set(B) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : ( ( A * set(A) ) > set(A) ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide,type,
    signed7115095781618012415divide: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
    signed6721504322012087516modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o * $o ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

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_Ohtt,type,
    time_htt: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) * nat ) > $o ) ).

tff(sy_c_Time__Reasoning_Otime,type,
    time_time: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > nat ) ).

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Ocontinuous__on,type,
    topolo81223032696312382ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

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

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

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

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
    topolo3814608138187158403Cauchy: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
    topolo6773858410816713723filter: 
      !>[A: $tType] : ( filter(A) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

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

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

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

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

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

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

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

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

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Type__Length_Olen0__class_Olen__of,type,
    type_len0_len_of: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_Typedef_Otype__definition,type,
    type_definition: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Uint32_OUint32,type,
    uint322: fun(code_integer,uint32) ).

tff(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: code_integer > uint32 ).

tff(sy_c_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__rel,type,
    vEBT_V2957053500504383685pi_rel: 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_Oreplicatei,type,
    vEBT_VEBT_replicatei: 
      !>[A: $tType] : ( ( nat * heap_Time_Heap(A) ) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

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_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__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_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__List__Assn_OlistI__assn,type,
    vEBT_List_listI_assn: 
      !>[A: $tType,B: $tType] : ( ( set(nat) * fun(A,fun(B,assn)) * list(A) * list(B) ) > assn ) ).

tff(sy_c_VEBT__List__Assn_Olist__assn,type,
    vEBT_List_list_assn: 
      !>[A: $tType,C: $tType] : ( ( fun(A,fun(C,assn)) * list(A) ) > fun(list(C),assn) ) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun(vEBT_VEBT,real) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Word_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 * word(A) ) > word(B) ) ).

tff(sy_c_Word_Oslice1,type,
    slice1: 
      !>[A: $tType,B: $tType] : ( ( nat * 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__bits__axioms,type,
    word_T1405001541757447525axioms: 
      !>[A: $tType,B: $tType] : ( ( fun(A,word(B)) * fun(nat,fun(A,A)) ) > $o ) ).

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_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

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

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_x____,type,
    x: list(vEBT_VEBTi) ).

tff(sy_v_xa____,type,
    xa: array(vEBT_VEBTi) ).

tff(sy_v_xb____,type,
    xb: vEBT_VEBTi ).

tff(sy_v_xc____,type,
    xc: vEBT_VEBTi ).

% Relevant facts (8896)
tff(fact_0_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: array(vEBT_VEBTi),Y14: vEBT_VEBTi] :
      ( ( vEBT_Nodei(X11,X12,X13,X14) = vEBT_Nodei(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_1_assnle,axiom,
    ! [TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] : entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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))) ).

% assnle
tff(fact_2_False,axiom,
    ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na))) ).

% False
tff(fact_3_div2__Suc__Suc,axiom,
    ! [M: 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,divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% div2_Suc_Suc
tff(fact_4_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),Q)
    <=> ( (B2)
       => entails(P,Q) ) ) ).

% ent_pure_pre_iff
tff(fact_5_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_6_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_7_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_8_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_9_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_10_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),divide_divide(A,B2,C2)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) ) ).

% times_divide_eq_left
tff(fact_11_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A2,B2),C2) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_12_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,A2,divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2) ) ).

% divide_divide_eq_right
tff(fact_13_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),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% times_divide_eq_right
tff(fact_14_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xd: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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),Xd),Xd)),Xd)),Xd) ) ).

% power4_eq_xxxx
tff(fact_15_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( pure_assn((P)) = pure_assn((Q)) )
    <=> ( (P)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_16_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_17_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_18_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_19_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_20_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_21_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_22_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xd: A,Y: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,Xd),Y)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) ) ) ).

% dvd_power_same
tff(fact_23_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),divide_divide(A,A2,B2)),Nb) = 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_24_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,K: num,L: num] : divide_divide(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = 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_25_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_26_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_27_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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_28_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_29_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_30_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(P,Q)
     => ( entails(Q,R)
       => entails(P,R) ) ) ).

% ent_trans
tff(fact_31_ent__refl,axiom,
    ! [P: assn] : entails(P,P) ).

% ent_refl
tff(fact_32_ent__iffI,axiom,
    ! [A3: assn,B4: assn] :
      ( entails(A3,B4)
     => ( entails(B4,A3)
       => ( A3 = B4 ) ) ) ).

% ent_iffI
tff(fact_33_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A2,B2),C2) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_34_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xd: A,Y: A,Z: A,W: A] : divide_divide(A,divide_divide(A,Xd,Y),divide_divide(A,Z,W)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),W),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_35_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xd: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,Xd,Y)),divide_divide(A,Z,W)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Z),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_36_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xd: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xd) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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),Xd),Nb)) ) ) ) ).

% power_commuting_commutes
tff(fact_37_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_38_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_39_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,A2,B2)),Nb) = 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_40_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_41_div__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : divide_divide(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) = divide_divide(nat,divide_divide(nat,M,Nb),Q2) ).

% div_mult2_eq
tff(fact_42_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q3: assn] :
      ( entails(P,P2)
     => ( entails(Q,Q3)
       => 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),P2),Q3)) ) ) ).

% ent_star_mono
tff(fact_43_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_44_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_45_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_46_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,$o),A3)) = A3 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
        <=> aa(A,$o,Q,X) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_48_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X: A] : aa(A,B,F2,X) = aa(A,B,G,X)
     => ( F2 = G ) ) ).

% HOL.ext
tff(fact_49_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : 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_50_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_51_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),divide_divide(A,B2,A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_52_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),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_53_div2__even__ext__nat,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( divide_divide(nat,Xd,aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,Y,aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xd)
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Y) )
       => ( Xd = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_54_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,divide_divide(A,B2,A2)),divide_divide(A,C2,A2))
            <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_55_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) )
            & ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = divide_divide(A,B2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_56_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_57_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one2 != bit0(Nb) ).

% semiring_norm(83)
tff(fact_58_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_59_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_60_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_61_pow__sum,axiom,
    ! [A2: nat,B2: 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_62_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xd: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xd) ).

% buildup_nothing_in_leaf
tff(fact_63_even__odd__cases,axiom,
    ! [Xd: nat] :
      ( ! [N: nat] : Xd != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)
     => ~ ! [N: nat] : Xd != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ).

% even_odd_cases
tff(fact_64_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_65_semiring__norm_I87_J,axiom,
    ! [M: num,Nb: num] :
      ( ( bit0(M) = bit0(Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(87)
tff(fact_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : vEBT_VEBT_bit_concat(H,L,D) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),D))),L) ).

% bit_concat_def
tff(fact_74_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_75_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_76_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_77_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_78_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)
           => ( 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ) ).

% div_add
tff(fact_79_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_80_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_81_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_82_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_83_add__self__div__2,axiom,
    ! [M: 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_84_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_85_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_86_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_87_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_88_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_89_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_90_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E: 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),E)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),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)),E)),C2) ) ).

% combine_common_factor
tff(fact_91_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_92_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_93_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_94_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_95_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ).

% add_divide_distrib
tff(fact_96_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_97_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)
         => ( 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_98_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)
         => ( 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_99_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_100_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_101_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_102_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_103_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_104_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,dvd_dvd(A,A2),A2) ) ).

% dvd_refl
tff(fact_105_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Nb: nat] : divide_divide(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)) = 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_106_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_107_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xd: 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),Xd),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),Xd),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))),Xd)),Y)) ) ).

% power2_sum
tff(fact_108_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ~ ! [K2: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_109_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_110_dvd__def,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) ) ) ).

% dvd_def
tff(fact_111_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_112_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_113_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_114_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_115_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,C2),D)
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ).

% mult_dvd_mono
tff(fact_116_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_117_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_118_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)
           => ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_119_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
         => ( aa(A,$o,dvd_dvd(A,C2),A2)
           => ( aa(A,$o,dvd_dvd(A,C2),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_120_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,D),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( divide_divide(A,divide_divide(A,A2,D),divide_divide(A,B2,D)) = divide_divide(A,A2,B2) ) ) ) ) ).

% div_div_div_same
tff(fact_121_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_122_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_123_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : divide_divide(A,A2,aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_124_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),divide_divide(A,B2,C2)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) ) ) ) ).

% dvd_div_mult
tff(fact_125_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),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ) ) ).

% div_mult_swap
tff(fact_126_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)
           => ( divide_divide(A,A2,divide_divide(A,B2,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_127_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)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_128_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),divide_divide(A,B2,C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_129_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,$o,dvd_dvd(A,D),C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),divide_divide(A,C2,D)) = 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),D)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_130_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_131_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_132_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xd,aa(num,A,numeral_numeral(A),bit0(one2)))),divide_divide(A,Xd,aa(num,A,numeral_numeral(A),bit0(one2)))) = Xd ) ).

% field_sum_of_halves
tff(fact_133_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,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))),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,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))),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,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))),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,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))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_134_triangle__def,axiom,
    ! [Nb: nat] : nat_triangle(Nb) = 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_135_high__def,axiom,
    ! [Xd: nat,Nb: nat] : vEBT_VEBT_high(Xd,Nb) = divide_divide(nat,Xd,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% high_def
tff(fact_136_bezout__lemma__nat,axiom,
    ! [D: nat,A2: nat,B2: nat,Xd: nat,Y: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D),A2)
     => ( aa(nat,$o,dvd_dvd(nat,D),B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D) ) )
         => ? [X: nat,Y2: nat] :
              ( aa(nat,$o,dvd_dvd(nat,D),A2)
              & aa(nat,$o,dvd_dvd(nat,D),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),X) = 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)),Y2)),D) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_137_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X: nat,Y2: 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),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),D2) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_138_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_139_highi__def,axiom,
    ! [Xd: nat,Nb: nat] : vEBT_VEBT_highi(Xd,Nb) = heap_Time_return(nat,divide_divide(nat,Xd,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% highi_def
tff(fact_140_set__decode__Suc,axiom,
    ! [Nb: nat,Xd: nat] :
      ( member(nat,aa(nat,nat,suc,Nb),nat_set_decode(Xd))
    <=> member(nat,Nb,nat_set_decode(divide_divide(nat,Xd,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_141_four__x__squared,axiom,
    ! [Xd: 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),Xd),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))),Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% four_x_squared
tff(fact_142_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_143_nat_Oinject,axiom,
    ! [X2: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y22) )
    <=> ( X2 = Y22 ) ) ).

% nat.inject
tff(fact_144_real__divide__square__eq,axiom,
    ! [R2: real,A2: real] : divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),R2)) = divide_divide(real,A2,R2) ).

% real_divide_square_eq
tff(fact_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_153_Suc__inject,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xd) = aa(nat,nat,suc,Y) )
     => ( Xd = Y ) ) ).

% Suc_inject
tff(fact_154_gcd__nat_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) )
     => ~ ( aa(nat,$o,dvd_dvd(nat,B2),A2)
          & ( B2 != A2 ) ) ) ).

% gcd_nat.asym
tff(fact_155_gcd__nat_Orefl,axiom,
    ! [A2: nat] : aa(nat,$o,dvd_dvd(nat,A2),A2) ).

% gcd_nat.refl
tff(fact_156_gcd__nat_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,A2),B2)
     => ( aa(nat,$o,dvd_dvd(nat,B2),C2)
       => aa(nat,$o,dvd_dvd(nat,A2),C2) ) ) ).

% gcd_nat.trans
tff(fact_157_gcd__nat_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & aa(nat,$o,dvd_dvd(nat,B2),A2) ) ) ).

% gcd_nat.eq_iff
tff(fact_158_gcd__nat_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,dvd_dvd(nat,A2),A2)
        & ( A2 != A2 ) ) ).

% gcd_nat.irrefl
tff(fact_159_gcd__nat_Oantisym,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,A2),B2)
     => ( aa(nat,$o,dvd_dvd(nat,B2),A2)
       => ( A2 = B2 ) ) ) ).

% gcd_nat.antisym
tff(fact_160_gcd__nat_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) )
     => ( ( aa(nat,$o,dvd_dvd(nat,B2),C2)
          & ( B2 != C2 ) )
       => ( aa(nat,$o,dvd_dvd(nat,A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_161_gcd__nat_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,A2),B2)
     => ( ( aa(nat,$o,dvd_dvd(nat,B2),C2)
          & ( B2 != C2 ) )
       => ( aa(nat,$o,dvd_dvd(nat,A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_162_gcd__nat_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) )
     => ( aa(nat,$o,dvd_dvd(nat,B2),C2)
       => ( aa(nat,$o,dvd_dvd(nat,A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_163_gcd__nat_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) )
    <=> ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ~ aa(nat,$o,dvd_dvd(nat,B2),A2) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_164_gcd__nat_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,A2),B2)
    <=> ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
          & ( A2 != B2 ) )
        | ( A2 = B2 ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_165_gcd__nat_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) )
    <=> ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_166_gcd__nat_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) )
     => aa(nat,$o,dvd_dvd(nat,A2),B2) ) ).

% gcd_nat.strict_implies_order
tff(fact_167_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,dvd_dvd(nat,A2),B2)
        & ( A2 != B2 ) )
     => ( A2 != B2 ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_168_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( aa(nat,$o,dvd_dvd(nat,A2),B2)
       => ( aa(nat,$o,dvd_dvd(nat,A2),B2)
          & ( A2 != B2 ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_169_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_170_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_171_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))
         => ? [B5: A,C3: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B5),C3) )
              & aa(A,$o,dvd_dvd(A,B5),B2)
              & aa(A,$o,dvd_dvd(A,C3),C2) ) ) ) ).

% division_decomp
tff(fact_172_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P3: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,P3),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ~ ! [X: A,Y2: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2) )
               => ( aa(A,$o,dvd_dvd(A,X),A2)
                 => ~ aa(A,$o,dvd_dvd(A,Y2),B2) ) ) ) ) ).

% dvd_productE
tff(fact_173_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_174_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_175_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_176_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_177_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_178_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_179_high__inv,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),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))),Xd),Nb) = Y ) ) ).

% high_inv
tff(fact_180_high__bound__aux,axiom,
    ! [Ma: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)))
     => aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Ma,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_181_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xd) = Y )
     => ( ( ( Xd = zero_zero(nat) )
         => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
         => ~ ! [N: nat] :
                ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_182_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_183_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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_184_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_185_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_186_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y22: num] :
      ( ( bit0(X2) = bit0(Y22) )
    <=> ( X2 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_187_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : divide_divide(int,aa(num,int,numeral_numeral(int),bit0(V)),aa(num,int,numeral_numeral(int),bit0(W))) = divide_divide(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_188_lowi__def,axiom,
    ! [Xd: nat,Nb: nat] : vEBT_VEBT_lowi(Xd,Nb) = heap_Time_return(nat,modulo_modulo(nat,Xd,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% lowi_def
tff(fact_189_list__assn__mono,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),P2: fun(A,fun(B,assn)),L: list(A),L2: list(B)] :
      ( ! [X: A,X3: B] : entails(aa(B,assn,aa(A,fun(B,assn),P,X),X3),aa(B,assn,aa(A,fun(B,assn),P2,X),X3))
     => entails(aa(list(B),assn,vEBT_List_list_assn(A,B,P,L),L2),aa(list(B),assn,vEBT_List_list_assn(A,B,P2,L),L2)) ) ).

% list_assn_mono
tff(fact_190_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xd: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb),Xd) ).

% buildup_nothing_in_min_max
tff(fact_191_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_192_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,modulo_modulo(A,A2,B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mod_trivial
tff(fact_193_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_194_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_195_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_196_pure__true,axiom,
    pure_assn($true) = one_one(assn) ).

% pure_true
tff(fact_197_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = one_one(assn) )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_198_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_199_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_200_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_201_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xd: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y) )
        <=> ( ( Xd = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_202_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xd: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y) = zero_zero(A) )
        <=> ( ( Xd = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% div_0
tff(fact_217_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_218_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_219_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_220_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_221_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( divide_divide(A,C2,A2) = divide_divide(A,C2,B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_222_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_223_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_224_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_225_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_226_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,one_one(A)) = A2 ) ).

% div_by_1
tff(fact_227_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,A2,one_one(A)) = A2 ) ).

% bits_div_by_1
tff(fact_228_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_229_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_230_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_231_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_232_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_233_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_234_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_235_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_236_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_237_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_238_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_239_less__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),one_one(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_one
tff(fact_240_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% neq0_conv
tff(fact_241_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_242_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self1
tff(fact_243_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self2
tff(fact_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_mod__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ( modulo_modulo(nat,M,Nb) = M ) ) ).

% mod_less
tff(fact_252_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q: assn] :
      ( entails(pure_assn((B2)),Q)
    <=> ( (B2)
       => entails(one_one(assn),Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_253_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_254_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_255_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_256_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_257_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_258_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_259_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_260_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xd: 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),Xd),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xd = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_261_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_262_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_263_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_264_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_265_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_266_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_267_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_268_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = divide_divide(A,one_one(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_269_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( divide_divide(A,one_one(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_270_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = divide_divide(A,B2,A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_271_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( divide_divide(A,B2,A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_272_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          divide_divide(A,A2,A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_273_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_274_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = divide_divide(A,A2,B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_275_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_276_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(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_277_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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_278_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: 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),divide_divide(A,A2,B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_279_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_280_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_281_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_282_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_283_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult1
tff(fact_284_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult2
tff(fact_285_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: 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),divide_divide(A,A2,B2)) ) ).

% div_mult_mult1_if
tff(fact_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_294_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_295_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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_296_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: 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_297_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_298_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_299_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_300_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_301_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_302_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_303_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: 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) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self4
tff(fact_304_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: 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) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self3
tff(fact_305_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: 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) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self2
tff(fact_306_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self1
tff(fact_307_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_308_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,divide_divide(A,one_one(A),A2)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_309_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,divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_310_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_311_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_312_div__by__Suc__0,axiom,
    ! [M: nat] : divide_divide(nat,M,aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_313_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_314_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_315_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_316_nat__power__eq__Suc__0__iff,axiom,
    ! [Xd: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xd),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( Xd = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_317_div__eq__dividend__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( ( divide_divide(nat,M,Nb) = M )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_318_div__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ( divide_divide(nat,M,Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_319_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_320_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_321_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_322_nat__zero__less__power__iff,axiom,
    ! [Xd: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xd),Nb))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Xd)
        | ( Nb = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_323_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_324_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_325_mod__by__Suc__0,axiom,
    ! [M: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_326_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: 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),divide_divide(nat,M,Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_327_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_328_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_329_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(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_330_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,$o,ord_less(A,divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_331_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,$o,ord_less(A,divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,ord_less(A,B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_332_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,$o,ord_less(A,one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,ord_less(A,B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_333_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,$o,ord_less(A,one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_334_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(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_335_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: 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_336_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = 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_337_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,ord_less(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_338_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),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_339_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self1
tff(fact_340_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self2
tff(fact_341_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self3
tff(fact_342_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self4
tff(fact_343_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_344_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_345_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_346_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xd: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B2)
         => ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xd)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,ord_less(nat,Xd),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_347_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_348_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),divide_divide(A,one_one(A),A2)) = divide_divide(A,B2,A2) ) ) ) ).

% unit_mult_div_div
tff(fact_349_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),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_350_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_351_div__mult__self__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( 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_352_div__mult__self1__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( 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_353_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(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_354_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Nb: 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) = modulo_modulo(nat,aa(nat,nat,suc,M),Nb) ).

% Suc_mod_mult_self1
tff(fact_355_Suc__mod__mult__self2,axiom,
    ! [M: nat,Nb: nat,K: 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) = modulo_modulo(nat,aa(nat,nat,suc,M),Nb) ).

% Suc_mod_mult_self2
tff(fact_356_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,M: 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) = modulo_modulo(nat,aa(nat,nat,suc,M),Nb) ).

% Suc_mod_mult_self3
tff(fact_357_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,M: 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) = modulo_modulo(nat,aa(nat,nat,suc,M),Nb) ).

% Suc_mod_mult_self4
tff(fact_358_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_359_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_360_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_361_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(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_362_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(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_363_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))),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_364_mod2__Suc__Suc,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% mod2_Suc_Suc
tff(fact_365_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_366_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_367_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_368_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(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_369_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_370_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: 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),Xd),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) )
        <=> ( ( Xd = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_371_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(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_372_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(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_373_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_374_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(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_375_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_376_add__self__mod__2,axiom,
    ! [M: 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_377_set__decode__0,axiom,
    ! [Xd: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xd))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xd) ) ).

% set_decode_0
tff(fact_378_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)
         => ( 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))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_379_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)
         => ( 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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_380_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)
         => ( 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))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_381_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_382_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_383_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_384_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_385_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)
           => ( 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)) = 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_386_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)
           => ( 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)),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_387_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( modulo_modulo(nat,M,D) = zero_zero(nat) )
     => ? [Q4: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D),Q4) ) ).

% mod_eq_0D
tff(fact_388_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_389_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_390_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ ! [C4: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) )
               => ( C4 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_391_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_392_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_393_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_394_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% gr0I
tff(fact_395_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_396_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),zero_zero(nat)) ).

% not_less0
tff(fact_397_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_398_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_399_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),Nb) ).

% less_not_refl
tff(fact_400_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Nb: nat] :
      ( ! [M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M2),zero_zero(nat))
     => ( ! [M2: nat,N: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),modulo_modulo(nat,M2,N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M2),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Nb) ) ) ).

% gcd_nat_induct
tff(fact_401_less__not__refl2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
     => ( M != Nb ) ) ).

% less_not_refl2
tff(fact_402_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( aa(nat,$o,ord_less(nat,S),T)
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_403_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,F2,Y3)),aa(A,B,F2,X))
                 => aa(A,$o,P,Y3) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_404_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_405_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_406_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,ord_less(nat,M3),N)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% nat_less_induct
tff(fact_407_infinite__descent,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M3: nat] :
              ( aa(nat,$o,ord_less(nat,M3),N)
              & ~ aa(nat,$o,P,M3) ) )
     => aa(nat,$o,P,Nb) ) ).

% infinite_descent
tff(fact_408_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M3: nat] :
                  ( aa(nat,$o,ord_less(nat,M3),N)
                  & ~ aa(nat,$o,P,M3) ) ) )
       => aa(nat,$o,P,Nb) ) ) ).

% infinite_descent0
tff(fact_409_linorder__neqE__nat,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( Xd != Y )
     => ( ~ aa(nat,$o,ord_less(nat,Xd),Y)
       => aa(nat,$o,ord_less(nat,Y),Xd) ) ) ).

% linorder_neqE_nat
tff(fact_410_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,F2,Y3)),aa(A,B,F2,X))
                 => aa(A,$o,P,Y3) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_411_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_412_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_413_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V2: fun(A,nat),Xd: A] :
      ( ! [X: A] :
          ( ~ aa(A,$o,P,X)
         => ? [Y3: A] :
              ( aa(nat,$o,ord_less(nat,aa(A,nat,V2,Y3)),aa(A,nat,V2,X))
              & ~ aa(A,$o,P,Y3) ) )
     => aa(A,$o,P,Xd) ) ).

% infinite_descent_measure
tff(fact_414_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,$o),Xd: A] :
      ( ! [X: A] :
          ( ( aa(A,nat,V2,X) = zero_zero(nat) )
         => aa(A,$o,P,X) )
     => ( ! [X: A] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,V2,X))
           => ( ~ aa(A,$o,P,X)
             => ? [Y3: A] :
                  ( aa(nat,$o,ord_less(nat,aa(A,nat,V2,Y3)),aa(A,nat,V2,X))
                  & ~ aa(A,$o,P,Y3) ) ) )
       => aa(A,$o,P,Xd) ) ) ).

% infinite_descent0_measure
tff(fact_415_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,ord_less(nat,Nb),N2)
         => ( 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),N2)) ) ) ) ).

% power_strict_increasing
tff(fact_416_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,ord_less(nat,Nb),N2)
         => ( 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),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_417_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_418_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_419_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_420_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_421_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(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_422_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,one_one(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_423_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [Y2: A] : aa(A,$o,ord_less(A,Y2),X4) ) ).

% linordered_field_no_lb
tff(fact_424_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [X_1: A] : aa(A,$o,ord_less(A,X4),X_1) ) ).

% linordered_field_no_ub
tff(fact_425_div__less__mono,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B4)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( ( modulo_modulo(nat,A3,Nb) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B4,Nb) = zero_zero(nat) )
           => aa(nat,$o,ord_less(nat,divide_divide(nat,A3,Nb)),divide_divide(nat,B4,Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_426_mod__less__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,ord_less(nat,modulo_modulo(nat,M,Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_427_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,divide_divide(nat,M,Nb)),M) ) ) ).

% div_less_dividend
tff(fact_428_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xd: A] :
          ( ( zero_zero(A) = Xd )
        <=> ( Xd = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_429_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xd: A] :
          ( ( one_one(A) = Xd )
        <=> ( Xd = one_one(A) ) ) ) ).

% one_reorient
tff(fact_430_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_431_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_432_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,modulo_modulo(A,A2,B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_433_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_434_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_435_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_436_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_437_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_438_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_439_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_440_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_441_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( ( Xd != Y )
         => ( ~ aa(A,$o,ord_less(A,Xd),Y)
           => aa(A,$o,ord_less(A,Y),Xd) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_442_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_443_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),modulo_modulo(nat,M,Nb))
    <=> ~ aa(nat,$o,dvd_dvd(nat,Nb),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_444_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_445_Suc__times__mod__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
     => ( 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_446_mod__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat,P3: nat,M: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,ord_less(nat,Nb),P3)
       => ( aa(nat,$o,ord_less(nat,M),P3)
         => ( ! [N: nat] :
                ( aa(nat,$o,ord_less(nat,N),P3)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,modulo_modulo(nat,aa(nat,nat,suc,N),P3)) ) )
           => aa(nat,$o,P,M) ) ) ) ) ).

% mod_induct
tff(fact_447_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_448_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_449_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,M: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less(A,aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(A,$o,ord_less(A,aa(nat,A,F2,Nb)),aa(nat,A,F2,M))
          <=> aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_450_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less(A,aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less(nat,Nb),N3)
           => aa(A,$o,ord_less(A,aa(nat,A,F2,Nb)),aa(nat,A,F2,N3)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_451_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% mod_0_imp_dvd
tff(fact_452_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_453_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_454_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_455_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_456_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_457_mod__Suc,axiom,
    ! [M: nat,Nb: nat] :
      modulo_modulo(nat,aa(nat,nat,suc,M),Nb) = $ite(aa(nat,nat,suc,modulo_modulo(nat,M,Nb)) = Nb,zero_zero(nat),aa(nat,nat,suc,modulo_modulo(nat,M,Nb))) ).

% mod_Suc
tff(fact_458_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_459_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_460_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Xd),zero_zero(A))
            | aa(A,$o,ord_less(A,Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_461_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_462_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_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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,divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_481_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,divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_482_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(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_483_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,divide_divide(A,A2,C2)),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_484_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(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_485_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,zero_zero(A)),divide_divide(A,Xd,Y)) ) ) ) ).

% divide_pos_pos
tff(fact_486_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less(A,divide_divide(A,Xd,Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_487_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,divide_divide(A,Xd,Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_488_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less(A,zero_zero(A)),divide_divide(A,Xd,Y)) ) ) ) ).

% divide_neg_neg
tff(fact_489_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_490_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb))
    <=> ( ( M = zero_zero(nat) )
        | ? [J2: nat] :
            ( ( M = aa(nat,nat,suc,J2) )
            & aa(nat,$o,ord_less(nat,J2),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_491_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_492_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_493_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
    <=> ? [M4: nat] : Nb = aa(nat,nat,suc,M4) ) ).

% gr0_conv_Suc
tff(fact_494_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_495_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( divide_divide(A,A2,B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_496_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_497_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_498_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_499_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_500_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Nb: 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_501_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_502_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_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D)) ) ) ) ).

% add_strict_mono
tff(fact_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_split__mod,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,modulo_modulo(nat,M,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,M) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I2: nat,J2: nat] :
              ( aa(nat,$o,ord_less(nat,J2),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)),J2) )
               => aa(nat,$o,P,J2) ) ) ) ) ) ).

% split_mod
tff(fact_526_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat,Xd: A] :
          ( ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
            | ( Xd = one_one(A) ) )
         => aa(A,$o,dvd_dvd(A,Xd),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)) ) ) ).

% dvd_power
tff(fact_527_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Nb)),aa(num,A,numeral_numeral(A),bit0(Q2))) = zero_zero(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_528_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : 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_529_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_530_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_531_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,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)
           => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_532_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,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_533_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),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_534_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_535_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_536_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xd: A,Y: A] : ~ aa(A,$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),Xd),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_537_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$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),Xd),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xd != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_538_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_539_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,ord_less(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_540_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),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_541_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,$o,ord_less(A,divide_divide(A,B2,C2)),A2)
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_542_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,$o,ord_less(A,A2),divide_divide(A,B2,C2))
          <=> aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_543_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,$o,ord_less(A,divide_divide(A,B2,C2)),A2)
          <=> aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_544_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,$o,ord_less(A,A2),divide_divide(A,B2,C2))
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_545_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xd: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less(A,Xd),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,ord_less(A,divide_divide(A,Xd,Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_546_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xd: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xd)
           => aa(A,$o,ord_less(A,Z),divide_divide(A,Xd,Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_547_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,divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_548_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,divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_549_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_550_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_551_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) )
             => ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C4) ) ) ) ).

% unit_dvdE
tff(fact_552_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))
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_553_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_554_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_555_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_556_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_557_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_558_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_eq
tff(fact_559_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A4,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B6),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_560_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_left_eq
tff(fact_561_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_right_eq
tff(fact_562_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_eq
tff(fact_563_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A4,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B6),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_564_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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),modulo_modulo(A,A2,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_565_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),modulo_modulo(A,A2,B2)) = 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_566_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_567_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_568_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( 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)) = divide_divide(nat,M,Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_569_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,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_570_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,Nb: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),modulo_modulo(A,A2,B2)),Nb),B2) = modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),B2) ) ).

% power_mod
tff(fact_571_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_572_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_573_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),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_574_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,$o,dvd_dvd(A,C2),modulo_modulo(A,A2,B2))
          <=> aa(A,$o,dvd_dvd(A,C2),A2) ) ) ) ).

% dvd_mod_iff
tff(fact_575_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( modulo_modulo(A,modulo_modulo(A,A2,B2),C2) = modulo_modulo(A,A2,C2) ) ) ) ).

% mod_mod_cancel
tff(fact_576_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),modulo_modulo(A,M,Nb)) ) ) ) ).

% dvd_mod
tff(fact_577_mod__Suc__eq,axiom,
    ! [M: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,M,Nb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,M),Nb) ).

% mod_Suc_eq
tff(fact_578_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,M,Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),Nb) ).

% mod_Suc_Suc_eq
tff(fact_579_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J3: nat] :
              ( aa(nat,$o,ord_less(nat,I),J3)
             => ( K != aa(nat,nat,suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_580_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_581_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,I)),K)
     => ~ ! [J3: nat] :
            ( aa(nat,$o,ord_less(nat,I),J3)
           => ( K != aa(nat,nat,suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_582_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_583_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_584_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_585_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_586_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_587_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_588_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_589_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_590_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_591_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_592_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_593_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,J3: nat,K2: nat] :
              ( aa(nat,$o,ord_less(nat,I3),J3)
             => ( aa(nat,$o,ord_less(nat,J3),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I3),J3)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J3),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I3),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_594_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_595_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_596_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_597_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_598_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_599_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_600_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_601_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_602_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_603_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_604_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Nb)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_605_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),one2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_606_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_607_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_608_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_609_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_610_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_611_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_612_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: 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_613_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)
           => ( 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)
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_614_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_615_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_616_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_617_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_618_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,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))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = divide_divide(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_619_list__decode_Ocases,axiom,
    ! [Xd: nat] :
      ( ( Xd != zero_zero(nat) )
     => ~ ! [N: nat] : Xd != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_620_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_621_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_622_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_623_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_624_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_625_nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_induct
tff(fact_626_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Nb: nat] :
      ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X),zero_zero(nat))
     => ( ! [Y2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y2))
       => ( ! [X: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X),Y2)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X)),aa(nat,nat,suc,Y2)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Nb) ) ) ) ).

% diff_induct
tff(fact_627_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_628_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_629_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_630_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_631_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% not0_implies_Suc
tff(fact_632_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_633_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_634_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_635_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_636_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_637_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A5: nat,B3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B3),A5) )
     => ( ! [A5: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A5),zero_zero(nat))
       => ( ! [A5: nat,B3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_638_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_639_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_640_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_641_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_642_nat__mod__eq__iff,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xd,Nb) = modulo_modulo(nat,Y,Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xd),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_643_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A2)
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_644_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( aa(nat,$o,dvd_dvd(nat,A2),zero_zero(nat))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_645_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A2)
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_646_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A2)
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_647_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,dvd_dvd(nat,A2),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_648_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(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_649_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)),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_650_pos2,axiom,
    aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_651_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))
                   => ( ( divide_divide(A,one_one(A),A2) = B3 )
                     => ( ( divide_divide(A,one_one(A),B3) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B3) = one_one(A) )
                         => ( divide_divide(A,C2,A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_652_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))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_653_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))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_654_split__div,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,divide_divide(nat,M,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I2: nat,J2: nat] :
              ( aa(nat,$o,ord_less(nat,J2),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)),J2) )
               => aa(nat,$o,P,I2) ) ) ) ) ) ).

% split_div
tff(fact_655_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),divide_divide(nat,M,Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_656_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),divide_divide(nat,M,Nb)))) ) ).

% dividend_less_times_div
tff(fact_657_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_658_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)
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_659_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)
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_660_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q2))) != 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_661_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) != 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_662_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D2: 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),D2)) ) ) ).

% mod_eqE
tff(fact_663_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2)) ) ).

% div_add1_eq
tff(fact_664_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)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_665_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(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_666_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_667_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_668_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_669_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_670_nat__induct2,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct2
tff(fact_671_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),divide_divide(nat,A3,Nb)),Nb)),modulo_modulo(nat,A3,Nb)) ).

% div_mod_decomp
tff(fact_672_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xd: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Xd)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xd)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_673_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_674_less__imp__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ? [K2: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_675_less__iff__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_iff_Suc_add
tff(fact_676_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_677_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_678_less__natE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ~ ! [Q4: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q4)) ) ).

% less_natE
tff(fact_679_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xd: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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),Xd),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_680_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_681_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,one_one(A),A2)),Nb) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_one_over
tff(fact_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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,$o,dvd_dvd(A,A2),divide_divide(A,C2,B2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_690_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,$o,dvd_dvd(A,divide_divide(A,A2,B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_691_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))
         => ( ( divide_divide(A,B2,A2) = divide_divide(A,C2,A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_692_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xd: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,Xd,Y) = divide_divide(A,W,Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_693_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: 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_694_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = 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_695_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) )
           => ( divide_divide(A,B2,C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_696_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 = divide_divide(A,B2,C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_697_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(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_698_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = 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_699_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_700_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,divide_divide(nat,M,Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_701_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_702_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_703_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_704_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_705_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_706_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_707_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_708_pow_Osimps_I1_J,axiom,
    ! [Xd: num] : pow(Xd,one2) = Xd ).

% pow.simps(1)
tff(fact_709_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] : ~ aa(A,$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),Xd),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_710_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$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),Xd),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)))))
        <=> ( ( Xd != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_711_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)),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_712_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)),divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_713_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) )
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_714_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_715_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_716_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),divide_divide(A,A2,B2))),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_717_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),divide_divide(A,A2,B2)),B2)),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_718_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),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) ) ).

% mod_div_decomp
tff(fact_719_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),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_720_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),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_721_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),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_722_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),divide_divide(A,A2,B2))),modulo_modulo(A,A2,B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_723_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: 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),divide_divide(A,B2,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2)) ) ).

% div_mult1_eq
tff(fact_724_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_725_mod__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q2: 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),modulo_modulo(nat,divide_divide(nat,M,Nb),Q2))),modulo_modulo(nat,M,Nb)) ).

% mod_mult2_eq
tff(fact_726_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_727_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_728_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_729_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_730_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_731_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_732_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_733_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: A,K: A,B2: A,A2: A] :
          ( ( B4 = 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),B4) = 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_734_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_735_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_736_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_737_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_738_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_739_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))
         => ( ( 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_740_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 = divide_divide(A,C2,B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_741_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)
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_742_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),divide_divide(A,A2,B2)),C2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2) ) ) ) ).

% unit_div_commute
tff(fact_743_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),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_744_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))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_745_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( 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_746_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) = 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_747_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),divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),B2,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_748_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),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_749_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xd: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xd,Y)),divide_divide(A,W,Z)) = 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),Xd),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_750_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xd: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xd,Y)),Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_frac_num
tff(fact_751_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xd: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,Xd,Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_num_frac
tff(fact_752_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),divide_divide(A,Y,Z)) = 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),Xd),Z)),Y),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_753_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xd,Z)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_754_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)
           => ( ( 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_755_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,$o,dvd_dvd(A,divide_divide(A,A2,B2)),C2)
            <=> aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_756_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,$o,dvd_dvd(A,A2),divide_divide(A,B2,C2))
            <=> aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_757_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,dvd_dvd(A,A2),B2)
             => ( aa(A,$o,dvd_dvd(A,C2),D)
               => ( ( divide_divide(A,B2,A2) = divide_divide(A,D,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),D) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_758_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_759_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D2: nat,X: nat,Y2: 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),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_760_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_761_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_762_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_763_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_764_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_765_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_766_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => aa(A,$o,ord_less(A,Xd),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_767_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_768_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_769_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat,M: nat] : modulo_modulo(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)) = divide_divide(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_770_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_771_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one2 != bit0(X2) ).

% verit_eq_simplify(10)
tff(fact_772_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)))
     => ( aa(nat,$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(Xd,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_773_nat__add__offset__less,axiom,
    ! [Y: nat,Nb: nat,Xd: 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,Xd),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),Xd),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_774_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,Xd: A,Nb: nat,Y: A] :
      ( ( replicate(A,M,Xd) = replicate(A,Nb,Y) )
    <=> ( ( M = Nb )
        & ( ( M != zero_zero(nat) )
         => ( Xd = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_775_nat__mod__eq_H,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,A2),Nb)
     => ( modulo_modulo(nat,A2,Nb) = A2 ) ) ).

% nat_mod_eq'
tff(fact_776_mod__lemma,axiom,
    ! [C2: nat,R2: nat,B2: nat,Q2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less(nat,R2),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),modulo_modulo(nat,Q2,C2))),R2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)) ) ) ).

% mod_lemma
tff(fact_777_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))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))))) ) ).

% even_even_mod_4_iff
tff(fact_778_msrevs_I1_J,axiom,
    ! [Nb: nat,K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( 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),divide_divide(nat,M,Nb)),K) ) ) ).

% msrevs(1)
tff(fact_779_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_780_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_781_not__real__square__gt__zero,axiom,
    ! [Xd: real] :
      ( ~ aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Xd))
    <=> ( Xd = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_782_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),A2)
     => ( ( divide_divide(int,A2,B2) = A2 )
      <=> ( B2 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_783_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_784_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less(num,M),one2) ).

% semiring_norm(75)
tff(fact_785_one__mod__exp__eq__one,axiom,
    ! [Nb: nat] : 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_786_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : 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))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_787_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa(num,$o,ord_less(num,one2),bit0(Nb)) ).

% semiring_norm(76)
tff(fact_788_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_789_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa(int,$o,ord_less(int,A3),B4)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
       => ( ( modulo_modulo(int,A3,Nb) = zero_zero(int) )
         => ( ( modulo_modulo(int,B4,Nb) = zero_zero(int) )
           => aa(int,$o,ord_less(int,divide_divide(int,A3,Nb)),divide_divide(int,B4,Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_790_real__arch__pow__inv,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => ? [N: nat] : aa(real,$o,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_791_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,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_792_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,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_793_int__div__less__self,axiom,
    ! [Xd: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less(int,one_one(int)),K)
       => aa(int,$o,ord_less(int,divide_divide(int,Xd,K)),Xd) ) ) ).

% int_div_less_self
tff(fact_794_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,divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_795_zdiv__mult__self,axiom,
    ! [M: int,A2: int,Nb: int] :
      ( ( M != zero_zero(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),divide_divide(int,A2,M)),Nb) ) ) ).

% zdiv_mult_self
tff(fact_796_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ? [R3: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),aa(nat,nat,suc,Nb)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_797_mod__exp__less__eq__exp,axiom,
    ! [A2: int,Nb: nat] : aa(int,$o,ord_less(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_798_pos__mod__bound2,axiom,
    ! [A2: int] : aa(int,$o,ord_less(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_799_axxmod2,axiom,
    ! [Xd: 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)),Xd)),Xd),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(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)),Xd)),Xd),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% axxmod2
tff(fact_800_nmod2,axiom,
    ! [Nb: int] :
      ( ( modulo_modulo(int,Nb,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) )
      | ( modulo_modulo(int,Nb,aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ).

% nmod2
tff(fact_801_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) != one_one(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_802_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),divide_divide(int,A3,Nb)),Nb)),modulo_modulo(int,A3,Nb)) ).

% div_mod_decomp_int
tff(fact_803_real__arch__pow,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => ? [N: nat] : aa(real,$o,ord_less(real,Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),N)) ) ).

% real_arch_pow
tff(fact_804_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)
       => ? [R3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),Nb) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_805_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)
       => ? [X: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),X)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X),Nb) = A2 )
            & ! [Y3: real] :
                ( ( aa(real,$o,ord_less(real,zero_zero(real)),Y3)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y3),Nb) = A2 ) )
               => ( Y3 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_806_p1mod22k_H,axiom,
    ! [B2: int,Nb: nat] : 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))),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_807_p1mod22k,axiom,
    ! [B2: int,Nb: nat] : 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))),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_808_z1pmod2,axiom,
    ! [B2: 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_809_axxdiv2,axiom,
    ! [Xd: 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)),Xd)),Xd),aa(num,int,numeral_numeral(int),bit0(one2))) = Xd )
      & ( 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)),Xd)),Xd),aa(num,int,numeral_numeral(int),bit0(one2))) = Xd ) ) ).

% axxdiv2
tff(fact_810_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => ( F3 = F3 ) ) ) ).

% minf(11)
tff(fact_811_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => ~ aa(A,$o,ord_less(A,T),X4) ) ) ).

% minf(7)
tff(fact_812_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => aa(A,$o,ord_less(A,X4),T) ) ) ).

% minf(5)
tff(fact_813_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => ( X4 != T ) ) ) ).

% minf(4)
tff(fact_814_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => ( X4 != T ) ) ) ).

% minf(3)
tff(fact_815_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X: A] :
              ( aa(A,$o,ord_less(A,X),Z3)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,ord_less(A,X),Z3)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,X4),Z2)
               => ( ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    | aa(A,$o,Q3,X4) ) ) ) ) ) ) ).

% minf(2)
tff(fact_816_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X: A] :
              ( aa(A,$o,ord_less(A,X),Z3)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,ord_less(A,X),Z3)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,X4),Z2)
               => ( ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    & aa(A,$o,Q3,X4) ) ) ) ) ) ) ).

% minf(1)
tff(fact_817_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => ( F3 = F3 ) ) ) ).

% pinf(11)
tff(fact_818_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => aa(A,$o,ord_less(A,T),X4) ) ) ).

% pinf(7)
tff(fact_819_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => ~ aa(A,$o,ord_less(A,X4),T) ) ) ).

% pinf(5)
tff(fact_820_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => ( X4 != T ) ) ) ).

% pinf(4)
tff(fact_821_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => ( X4 != T ) ) ) ).

% pinf(3)
tff(fact_822_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X: A] :
              ( aa(A,$o,ord_less(A,Z3),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,ord_less(A,Z3),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,Z2),X4)
               => ( ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    | aa(A,$o,Q3,X4) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_823_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X: A] :
              ( aa(A,$o,ord_less(A,Z3),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,ord_less(A,Z3),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,Z2),X4)
               => ( ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    & aa(A,$o,Q3,X4) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_824_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,
    ! [Xd: nat] :
      ( ( Xd != zero_zero(nat) )
     => ( ( Xd != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : Xd != aa(nat,nat,suc,aa(nat,nat,suc,Va)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_825_nat__mod__eq,axiom,
    ! [B2: nat,Nb: nat,A2: nat] :
      ( aa(nat,$o,ord_less(nat,B2),Nb)
     => ( ( modulo_modulo(nat,A2,Nb) = modulo_modulo(nat,B2,Nb) )
       => ( modulo_modulo(nat,A2,Nb) = B2 ) ) ) ).

% nat_mod_eq
tff(fact_826_mod__plus__right,axiom,
    ! [A2: nat,Xd: nat,M: nat,B2: nat] :
      ( ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Xd),M) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Xd),M) )
    <=> ( modulo_modulo(nat,A2,M) = modulo_modulo(nat,B2,M) ) ) ).

% mod_plus_right
tff(fact_827_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))),aa(A,A,aa(nat,fun(A,A),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_828_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% pinf(9)
tff(fact_829_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% pinf(10)
tff(fact_830_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% minf(9)
tff(fact_831_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% minf(10)
tff(fact_832_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)
      <=> ( modulo_modulo(nat,B2,Nb) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_833_msrevs_I2_J,axiom,
    ! [K: nat,Nb: nat,M: 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) = modulo_modulo(nat,M,Nb) ).

% msrevs(2)
tff(fact_834_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(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,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Nb),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_835_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X5: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X5))
        <=> ? [X5: A] :
              ( aa(A,$o,dvd_dvd(A,L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),zero_zero(A)))
              & aa(A,$o,P,X5) ) ) ) ).

% unity_coeff_ex
tff(fact_836_z1pdiv2,axiom,
    ! [B2: 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_837_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,divide_divide(nat,A2,C2)),B2) ) ) ).

% td_gal_lt
tff(fact_838_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_839_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_840_low__inv,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),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))),Xd),Nb) = Xd ) ) ).

% low_inv
tff(fact_841_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),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,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_842_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_843_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(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,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Nb),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_844_low__def,axiom,
    ! [Xd: nat,Nb: nat] : vEBT_VEBT_low(Xd,Nb) = modulo_modulo(nat,Xd,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% low_def
tff(fact_845_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: 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),D),modulo_modulo(nat,K,L)) )
     => ( aa(nat,$o,ord_less(nat,Nb),L)
       => ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D),Nb),L) = Nb ) ) ) ).

% word_rot_lem
tff(fact_846_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_847_mlex__snd__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B6: nat,N2: nat] :
      ( ( A2 = A4 )
     => ( aa(nat,$o,ord_less(nat,B2),B6)
       => aa(nat,$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),N2)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N2)),B6)) ) ) ).

% mlex_snd_decrI
tff(fact_848_bit__split__inv,axiom,
    ! [Xd: nat,D: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xd,D),vEBT_VEBT_low(Xd,D),D) = Xd ).

% bit_split_inv
tff(fact_849_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),Nb)
    <=> ( Nb != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_850_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_851_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se8732182000553998342ip_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_852_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_853_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_854_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_855_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_856_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_857_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo(int,M,D) = zero_zero(int) )
     => ? [Q4: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q4) ) ).

% zmod_eq_0D
tff(fact_858_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_859_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,M),T)
      <=> aa(int,$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),T)) ) ) ).

% zdvd_mono
tff(fact_860_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo(int,M,D) = zero_zero(int) )
    <=> ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q5) ) ).

% zmod_eq_0_iff
tff(fact_861_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_862_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_863_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_864_zmod__helper,axiom,
    ! [Nb: int,M: int,K: int,A2: int] :
      ( ( modulo_modulo(int,Nb,M) = K )
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),A2),M) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),A2),M) ) ) ).

% zmod_helper
tff(fact_865_mod__plus__cong,axiom,
    ! [B2: int,B6: int,Xd: int,X6: int,Y: int,Y4: int,Z4: int] :
      ( ( B2 = B6 )
     => ( ( modulo_modulo(int,Xd,B6) = modulo_modulo(int,X6,B6) )
       => ( ( modulo_modulo(int,Y,B6) = modulo_modulo(int,Y4,B6) )
         => ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X6),Y4) = Z4 )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),Y),B2) = modulo_modulo(int,Z4,B6) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_866_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_867_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_868_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_869_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_870_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_871_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_872_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),modulo_modulo(int,K,L)) ) ).

% neg_mod_bound
tff(fact_873_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,modulo_modulo(int,K,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_874_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M3: extended_enat] :
              ( aa(extended_enat,$o,ord_less(extended_enat,M3),N)
             => aa(extended_enat,$o,P,M3) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Nb) ) ).

% enat_less_induct
tff(fact_875_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa(extended_enat,$o,ord_less(extended_enat,Nb),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_876_less__int__code_I1_J,axiom,
    ~ aa(int,$o,ord_less(int,zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_877_zdvd__not__zless,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),M)
     => ( aa(int,$o,ord_less(int,M),Nb)
       => ~ aa(int,$o,dvd_dvd(int,Nb),M) ) ) ).

% zdvd_not_zless
tff(fact_878_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xd: A] : neg_numeral_dbl(A,Xd) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Xd) ) ).

% dbl_def
tff(fact_879_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)))
     => ( aa(nat,$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(Xd,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_880_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))),aa(A,A,aa(nat,fun(A,A),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_881_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_882_mlex__bound,axiom,
    ! [A2: nat,A3: nat,B2: nat,N2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),A3)
     => ( aa(nat,$o,ord_less(nat,B2),N2)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N2)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)) ) ) ).

% mlex_bound
tff(fact_883_mlex__fst__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,N2: nat,B6: nat] :
      ( aa(nat,$o,ord_less(nat,A2),A4)
     => ( aa(nat,$o,ord_less(nat,B2),N2)
       => ( aa(nat,$o,ord_less(nat,B6),N2)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N2)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N2)),B6)) ) ) ) ).

% mlex_fst_decrI
tff(fact_884_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xd: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Xd))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),Xd)
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_885_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_886_forall__finite_I3_J,axiom,
    ! [Xd: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,aa(nat,nat,suc,Xd)))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Xd))
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% forall_finite(3)
tff(fact_887_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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_888_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_889_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C2: A,D: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_890_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_891_div__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : divide_divide(word(A),one_one(word(A)),W) = aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))) ) ).

% div_word_one
tff(fact_892_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),one_one(word(A)))
        <=> ( Xd = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_Suc__0__mod__eq,axiom,
    ! [Nb: 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_901_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P3: $o] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> (P3) ) ) ).

% odd_of_bool_self
tff(fact_902_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : divide_divide(A,aa($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_903_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_904_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_905_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($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_906_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_907_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_908_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)
         => ? [N: word(A)] : M = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))) ) ) ).

% word_gr0_conv_Suc
tff(fact_909_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A))))
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_910_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_911_word__induct__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( aa(word(A),$o,ord_less(word(A),N),M)
               => ( aa(word(A),$o,P,N)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) ) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct_less
tff(fact_912_word__induct2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),Nb: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N) != zero_zero(word(A)) )
               => ( aa(word(A),$o,P,N)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) ) )
           => aa(word(A),$o,P,Nb) ) ) ) ).

% word_induct2
tff(fact_913_word__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( aa(word(A),$o,P,N)
               => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct
tff(fact_914_div__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( divide_divide(word(A),W,W) = one_one(word(A)) ) ) ) ).

% div_word_self
tff(fact_915_word__div__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : divide_divide(word(A),Nb,one_one(word(A))) = Nb ) ).

% word_div_1
tff(fact_916_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: word(A)] :
          ( ( Xd != zero_zero(word(A)) )
         => ( ( Nb != one_one(word(A)) )
           => aa(word(A),$o,ord_less(word(A),divide_divide(word(A),Xd,Nb)),Xd) ) ) ) ).

% div_less_dividend_word
tff(fact_917_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_918_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o,Q2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P3)) = aa($o,A,zero_neq_one_of_bool(A),(Q2)) )
        <=> ( (P3)
          <=> (Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_919_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_920_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ~ ( ( (P3)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P3)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_921_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ( ( (P3)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P3)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_922_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P3)) = $ite((P3),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_923_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)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_924_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A5: A] :
              ( ( divide_divide(A,A5,aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
             => aa(A,$o,P,A5) )
         => ( ! [A5: A,B3: $o] :
                ( aa(A,$o,P,A5)
               => ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5)),aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5))) ) )
           => aa(A,$o,P,A2) ) ) ) ).

% bits_induct
tff(fact_925_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($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_926_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_927_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D ) )
        <=> ( 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),D)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_928_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,Xd: 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),Xd),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),Xd),Y)) )
        <=> ( ( W = Xd )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_929_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_930_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_931_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,ord_less(A,Xd),Y) ) ) ) ).

% mult_less_iff1
tff(fact_932_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => ( aa(A,$o,ord_less_eq(A,B2),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))),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)) = divide_divide(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_933_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$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,$o,ord_less_eq(A,A2),zero_zero(A)) )
              | ( aa(nat,$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_934_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),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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_935_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_936_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,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)))),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_937_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xd: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
             => aa(A,$o,ord_less_eq(A,U),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_938_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: option(A)] : ~ aa(option(A),$o,ord_less(option(A),Xd),none(A)) ) ).

% less_option_None
tff(fact_939_min__in__set__def,axiom,
    ! [Xs: set(nat),Xd: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xd)
    <=> ( member(nat,Xd,Xs)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => aa(nat,$o,ord_less_eq(nat,Xd),X5) ) ) ) ).

% min_in_set_def
tff(fact_940_max__in__set__def,axiom,
    ! [Xs: set(nat),Xd: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xd)
    <=> ( member(nat,Xd,Xs)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => aa(nat,$o,ord_less_eq(nat,X5),Xd) ) ) ) ).

% max_in_set_def
tff(fact_941_semiring__norm_I71_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,ord_less_eq(num,bit0(M)),bit0(Nb))
    <=> aa(num,$o,ord_less_eq(num,M),Nb) ) ).

% semiring_norm(71)
tff(fact_942_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa(num,$o,ord_less_eq(num,one2),Nb) ).

% semiring_norm(68)
tff(fact_943_Suc__le__mono,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Nb)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,ord_less_eq(nat,Nb),M) ) ).

% Suc_le_mono
tff(fact_944_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_945_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),Nb) ).

% le0
tff(fact_946_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_947_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: option(A)] : aa(option(A),$o,ord_less_eq(option(A),none(A)),Xd) ) ).

% less_eq_option_None_code
tff(fact_948_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),zero_zero(word(A)))
        <=> ( Xd = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_949_mod__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(word(A),W,W) = zero_zero(word(A)) ) ).

% mod_word_self
tff(fact_950_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A2),zero_zero(word(A)))
        <=> ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_951_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,ord_less_eq(A,Nb),zero_zero(A))
        <=> ( Nb = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_952_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,ord_less_eq(num,M),Nb) ) ) ).

% numeral_le_iff
tff(fact_953_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_954_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_955_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less_eq(num,bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_956_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_957_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,ord_less_eq(A,aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_958_div__of__0__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : divide_divide(word(A),zero_zero(word(A)),Nb) = zero_zero(word(A)) ) ).

% div_of_0_id
tff(fact_959_enat__ord__number_I1_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(extended_enat,$o,ord_less_eq(extended_enat,aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$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_960_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_961_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_962_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_963_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_964_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_965_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_966_one__le__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb))
    <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),Nb) ) ) ).

% one_le_mult_iff
tff(fact_967_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ) ).

% mult_le_cancel2
tff(fact_968_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_969_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_970_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_971_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_972_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_973_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se8732182000553998342ip_bit(int,Nb,K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_974_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Nb)),one_one(A))
        <=> aa(num,$o,ord_less_eq(num,Nb),one2) ) ) ).

% numeral_le_one_iff
tff(fact_975_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,one_one(A),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_976_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),divide_divide(A,one_one(A),A2))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_977_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xd: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B2)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xd)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,ord_less_eq(nat,Xd),Y) ) ) ) ).

% power_increasing_iff
tff(fact_978_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(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_979_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_980_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,ord_less_eq(int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_981_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ( aa(int,$o,ord_less(int,L),K)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_982_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,K),L)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_983_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,K),L)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_984_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ( aa(int,$o,ord_less(int,L),K)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_985_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,$o,ord_less_eq(A,divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,ord_less_eq(A,A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_986_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,$o,ord_less_eq(A,divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_987_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,$o,ord_less_eq(A,one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_988_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,$o,ord_less_eq(A,one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,ord_less_eq(A,A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_989_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,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,ord_less_eq(nat,Nb),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_990_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
             => ( aa(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,ord_less_eq(A,A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_991_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_992_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$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_993_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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))) )
            <=> ( Xd = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_994_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_995_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$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,ord_less_eq(A,zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_996_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),divide_divide(word(A),K,Xd))
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xd)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xd)),K) ) ) ) ).

% div_le_mult
tff(fact_997_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),divide_divide(word(A),K,Xd))
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xd)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xd)),K) ) ) ) ).

% div_lt_mult
tff(fact_998_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),divide_divide(word(A),A2,C2)),B2) ) ) ) ).

% More_Word.word_div_mult
tff(fact_999_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)),divide_divide(word(A),Nb,B2)),B2)),modulo_modulo(word(A),Nb,B2)) = Nb ) ).

% word_mod_div_equality
tff(fact_1000_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Z))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Z)) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_1001_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y))
         => ( ( Xd != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_1002_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,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_1003_div__by__0__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : divide_divide(word(A),Xd,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% div_by_0_word
tff(fact_1004_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)
         => ( divide_divide(word(A),W,V) = zero_zero(word(A)) ) ) ) ).

% word_div_less
tff(fact_1005_word__less__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ( divide_divide(word(A),Xd,Y) = zero_zero(word(A)) )
         => ( ( Y = zero_zero(word(A)) )
            | aa(word(A),$o,ord_less(word(A),Xd),Y) ) ) ) ).

% word_less_div
tff(fact_1006_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),Y)
         => ( divide_divide(word(A),Xd,Y) = zero_zero(word(A)) ) ) ) ).

% word_div_lt_eq_0
tff(fact_1007_word__mod__less__divisor,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),modulo_modulo(word(A),M,Nb)),Nb) ) ) ).

% word_mod_less_divisor
tff(fact_1008_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),A2)
        <=> ( A2 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_1009_word__le__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),B2),A2)
        <=> ~ aa(word(A),$o,ord_less(word(A),A2),B2) ) ) ).

% word_le_not_less
tff(fact_1010_word__gt__a__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),Nb)
         => aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Nb) ) ) ).

% word_gt_a_gt_0
tff(fact_1011_word__mod__by__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] : modulo_modulo(word(A),K,zero_zero(word(A))) = K ) ).

% word_mod_by_0
tff(fact_1012_word__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
        <=> aa(word(A),$o,ord_less(word(A),Y),Xd) ) ) ).

% word_not_le
tff(fact_1013_word__neq__0__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
        <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),W) ) ) ).

% word_neq_0_conv
tff(fact_1014_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),Y) ) ).

% word_zero_le
tff(fact_1015_word__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Y)
        <=> ( zero_zero(word(A)) != Y ) ) ) ).

% word_gt_0
tff(fact_1016_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A2),zero_zero(word(A)))
         => ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_1017_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,ord_less(word(A),A2),zero_zero(word(A))) ) ).

% word_coorder.extremum_strict
tff(fact_1018_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),A2) ) ).

% word_coorder.extremum
tff(fact_1019_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : ~ aa(word(A),$o,ord_less(word(A),Xd),zero_zero(word(A))) ) ).

% word_not_simps(1)
tff(fact_1020_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less_eq(nat,Nb),N3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,F2,Nb)),aa(nat,A,F2,N3)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_1021_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,N))
         => ( aa(nat,$o,ord_less_eq(nat,Nb),N3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,F2,N3)),aa(nat,A,F2,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_1022_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : aa(A,$o,ord_less_eq(A,A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_1023_verit__la__generic,axiom,
    ! [A2: int,Xd: int] :
      ( aa(int,$o,ord_less_eq(int,A2),Xd)
      | ( A2 = Xd )
      | aa(int,$o,ord_less_eq(int,Xd),A2) ) ).

% verit_la_generic
tff(fact_1024_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ aa(A,$o,ord_less_eq(A,A2),B2)
          | ~ aa(A,$o,ord_less_eq(A,B2),A2) ) ) ).

% verit_la_disequality
tff(fact_1025_mod__word__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),V)
         => ( modulo_modulo(word(A),W,V) = W ) ) ) ).

% mod_word_less
tff(fact_1026_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
        <=> ( ( Xd = Y )
            | aa(word(A),$o,ord_less(word(A),Xd),Y) ) ) ) ).

% word_le_less_eq
tff(fact_1027_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y4: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y4)),Xd)
         => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y)),Xd)
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y4)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y))
            <=> aa(word(A),$o,ord_less(word(A),Y4),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_1028_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y4: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y4))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y))
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y4)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y))
            <=> aa(word(A),$o,ord_less(word(A),Y4),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_1029_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: option(A)] : aa(option(A),$o,ord_less_eq(option(A),none(A)),Xd) ) ).

% less_eq_option_None
tff(fact_1030_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: option(A)] :
          ( aa(option(A),$o,ord_less_eq(option(A),Xd),none(A))
         => ( Xd = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_1031_subset__decode__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),nat_set_decode(M)),nat_set_decode(Nb))
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% subset_decode_imp_le
tff(fact_1032_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,ord_less_eq(nat,Nb),N2)
         => ( aa(A,$o,ord_less_eq(A,one_one(A)),A2)
           => aa(A,$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),N2)) ) ) ) ).

% power_increasing
tff(fact_1033_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_1034_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_1035_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xd: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd) ) ).

% zero_le
tff(fact_1036_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_1037_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A4: A] :
          ( ~ aa(A,$o,ord_less_eq(A,B6),A4)
        <=> aa(A,$o,ord_less(A,A4),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_1038_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => ~ aa(A,$o,ord_less_eq(A,X4),T) ) ) ).

% pinf(6)
tff(fact_1039_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,Z2),X4)
         => aa(A,$o,ord_less_eq(A,T),X4) ) ) ).

% pinf(8)
tff(fact_1040_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => aa(A,$o,ord_less_eq(A,X4),T) ) ) ).

% minf(6)
tff(fact_1041_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( aa(A,$o,ord_less(A,X4),Z2)
         => ~ aa(A,$o,ord_less_eq(A,T),X4) ) ) ).

% minf(8)
tff(fact_1042_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_1043_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_1044_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
        <=> ? [C5: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) ) ) ).

% le_iff_add
tff(fact_1045_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => aa(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_1046_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ~ ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% less_eqE
tff(fact_1047_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => aa(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_1048_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D)) ) ) ) ).

% add_mono
tff(fact_1049_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1050_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1051_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & ( K = L ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1052_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_1053_le__num__One__iff,axiom,
    ! [Xd: num] :
      ( aa(num,$o,ord_less_eq(num,Xd),one2)
    <=> ( Xd = one2 ) ) ).

% le_num_One_iff
tff(fact_1054_Suc__leD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),Nb)
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% Suc_leD
tff(fact_1055_le__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,ord_less_eq(nat,M),Nb)
       => ( M = aa(nat,nat,suc,Nb) ) ) ) ).

% le_SucE
tff(fact_1056_le__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Nb)) ) ).

% le_SucI
tff(fact_1057_Suc__le__D,axiom,
    ! [Nb: nat,M6: nat] :
      ( aa(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_1058_le__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,ord_less_eq(nat,M),Nb)
        | ( M = aa(nat,nat,suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_1059_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_1060_not__less__eq__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,ord_less_eq(nat,M),Nb)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Nb)),M) ) ).

% not_less_eq_eq
tff(fact_1061_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M3)),N)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% full_nat_induct
tff(fact_1062_nat__induct__at__least,axiom,
    ! [M: nat,Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( aa(nat,$o,P,M)
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_1063_transitive__stepwise__le,axiom,
    ! [M: nat,Nb: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X),X)
       => ( ! [X: nat,Y2: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X),Y2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y2),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X),Z2) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,M),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1064_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),Nb) ).

% less_eq_nat.simps(1)
tff(fact_1065_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1066_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1067_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Nb),zero_zero(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_1068_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,$o,ord_less_eq(nat,N5),Nb)
          & aa(nat,$o,P,N5) ) ) ).

% exists_leI
tff(fact_1069_nat__less__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
    <=> ( aa(nat,$o,ord_less_eq(nat,M),Nb)
        & ( M != Nb ) ) ) ).

% nat_less_le
tff(fact_1070_less__imp__le__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% less_imp_le_nat
tff(fact_1071_le__eq__less__or__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1072_less__or__eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,$o,ord_less(nat,M),Nb)
        | ( M = Nb ) )
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% less_or_eq_imp_le
tff(fact_1073_le__neq__implies__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( ( M != Nb )
       => aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% le_neq_implies_less
tff(fact_1074_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),J3)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,F2,I3)),aa(nat,nat,F2,J3)) )
     => ( aa(nat,$o,ord_less_eq(nat,I),J)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,F2,I)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_1075_nat__le__iff__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
    <=> ? [K3: nat] : Nb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_1076_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_1077_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_1078_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_1079_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,K),L)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_1080_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),L)
     => ? [N: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) ) ).

% le_Suc_ex
tff(fact_1081_add__leD2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => aa(nat,$o,ord_less_eq(nat,K),Nb) ) ).

% add_leD2
tff(fact_1082_add__leD1,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% add_leD1
tff(fact_1083_le__add2,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) ).

% le_add2
tff(fact_1084_le__add1,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)) ).

% le_add1
tff(fact_1085_add__leE,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => ~ ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ~ aa(nat,$o,ord_less_eq(nat,K),Nb) ) ) ).

% add_leE
tff(fact_1086_less__eq__real__def,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,Xd),Y)
    <=> ( aa(real,$o,ord_less(real,Xd),Y)
        | ( Xd = Y ) ) ) ).

% less_eq_real_def
tff(fact_1087_conj__le__cong,axiom,
    ! [Xd: int,X6: int,P: $o,P2: $o] :
      ( ( Xd = X6 )
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
            & (P) )
        <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_1088_imp__le__cong,axiom,
    ! [Xd: int,X6: int,P: $o,P2: $o] :
      ( ( Xd = X6 )
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
           => (P) )
        <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_1089_less__eq__int__code_I1_J,axiom,
    aa(int,$o,ord_less_eq(int,zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_1090_div__le__mono,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => aa(nat,$o,ord_less_eq(nat,divide_divide(nat,M,K)),divide_divide(nat,Nb,K)) ) ).

% div_le_mono
tff(fact_1091_div__le__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,ord_less_eq(nat,divide_divide(nat,M,Nb)),M) ).

% div_le_dividend
tff(fact_1092_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_cube
tff(fact_1093_le__square,axiom,
    ! [M: nat] : aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_1094_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,K),L)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_1095_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_1096_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_1097_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),Xd)
        <=> ( Xd != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_1098_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),I)
         => ( aa(word(A),$o,ord_less(word(A),I),M)
           => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A))))
              & aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ) ) ).

% inc_i
tff(fact_1099_inc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),M)
         => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ).

% inc_le
tff(fact_1100_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,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_1101_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),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,ord_less_eq(word(A),Xd),Nb) ) ) ).

% plus_one_helper
tff(fact_1102_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),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),Xd),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_1103_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_1104_mod__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,ord_less_eq(nat,modulo_modulo(nat,M,Nb)),M) ).

% mod_less_eq_dividend
tff(fact_1105_i0__lb,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,$o,ord_less_eq(extended_enat,zero_zero(extended_enat)),Nb) ).

% i0_lb
tff(fact_1106_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,ord_less_eq(extended_enat,Nb),zero_zero(extended_enat))
    <=> ( Nb = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_1107_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,ord_less_eq(nat,Nb),N2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
           => ( aa(A,$o,ord_less_eq(A,A2),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_1108_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,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1109_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,ord_less_eq(int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_1110_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa(int,$o,ord_less_eq(int,K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K)) ).

% set_bit_greater_eq
tff(fact_1111_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,ord_less_eq(A,Xd),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1112_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,ord_less_eq(A,Xd),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1113_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_1114_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_le_numeral
tff(fact_1115_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y) = zero_zero(A) )
            <=> ( ( Xd = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1116_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y) = zero_zero(A) )
            <=> ( ( Xd = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1117_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B2),zero_zero(A))
           => aa(A,$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_1118_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(A,$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_1119_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,B2),A2)
           => aa(A,$o,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing2
tff(fact_1120_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,A2),B2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_1121_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,B2),C2)
           => aa(A,$o,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing
tff(fact_1122_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,C2),B2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_1123_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_1124_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1125_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less_eq(A,one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_1126_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ) ).

% mult_mono
tff(fact_1127_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1128_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : aa(A,$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_1129_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) )
            | ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) ) )
         => aa(A,$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_1130_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B2),A2)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1131_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B2),zero_zero(A))
           => aa(A,$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_1132_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_1133_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B2),A2)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1134_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_1135_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1136_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) ) )
         => aa(A,$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_1137_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(A,$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_1138_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,B2),zero_zero(A))
           => aa(A,$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_1139_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(A,$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_1140_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,B2),zero_zero(A))
           => aa(A,$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_1141_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) )
            | ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1142_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1143_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D)) ) ) ) ).

% add_less_le_mono
tff(fact_1144_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D)) ) ) ) ).

% add_le_less_mono
tff(fact_1145_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less(A,I),J)
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1146_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & aa(A,$o,ord_less(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1147_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,ord_less_eq(A,one_one(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_le_numeral
tff(fact_1148_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,divide_divide(A,B2,C2)),divide_divide(A,A2,C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1149_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),divide_divide(A,Xd,Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1150_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,divide_divide(A,Xd,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1151_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,divide_divide(A,Xd,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1152_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),divide_divide(A,Xd,Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1153_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),divide_divide(A,A2,B2))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) )
            | ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1154_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_right_mono
tff(fact_1155_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1156_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => aa(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_1157_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
           => aa(A,$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_1158_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),A2)
         => aa(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_1159_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => aa(A,$o,ord_less_eq(A,modulo_modulo(A,A2,B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1160_nat__compl__induct_H,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nn),N)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_compl_induct'
tff(fact_1161_nat__compl__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nn),N)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_compl_induct
tff(fact_1162_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,ord_less(nat,A2),B2)
        & aa(nat,$o,ord_less_eq(nat,B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = aa(nat,nat,suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_1163_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(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_1164_le__imp__less__Suc,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1165_less__eq__Suc__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Nb)),M) ) ).

% less_eq_Suc_le
tff(fact_1166_less__Suc__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% less_Suc_eq_le
tff(fact_1167_le__less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1168_Suc__le__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1169_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I),N)
             => ( aa(nat,$o,ord_less(nat,N),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N))
                 => aa(nat,$o,P,N) ) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% inc_induct
tff(fact_1170_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I),N)
             => ( aa(nat,$o,ord_less(nat,N),J)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_1171_Suc__le__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1172_Suc__leI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),Nb) ) ).

% Suc_leI
tff(fact_1173_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,ord_less_eq(nat,K2),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,ord_less(nat,I4),K2)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_1174_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa(nat,$o,ord_less(nat,M2),N)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,F2,M2)),aa(nat,nat,F2,N)) )
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,M)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))) ) ).

% mono_nat_linear_lb
tff(fact_1175_nat__geq__1__eq__neqz,axiom,
    ! [Xd: nat] :
      ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Xd)
    <=> ( Xd != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1176_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$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_1177_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,$o,ord_less_eq(nat,M),Nb)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2) ) ) ) ).

% power_le_dvd
tff(fact_1178_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xd: A,Y: A,Nb: nat,M: nat] :
          ( aa(A,$o,dvd_dvd(A,Xd),Y)
         => ( aa(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),Xd),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_1179_Suc__div__le__mono,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,ord_less_eq(nat,divide_divide(nat,M,Nb)),divide_divide(nat,aa(nat,nat,suc,M),Nb)) ).

% Suc_div_le_mono
tff(fact_1180_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_1181_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_1182_mlex__leI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B6: nat,N2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A2),A4)
     => ( aa(nat,$o,ord_less_eq(nat,B2),B6)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N2)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N2)),B6)) ) ) ).

% mlex_leI
tff(fact_1183_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,ord_less_eq(A,aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_1184_mod__Suc__le__divisor,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,ord_less_eq(nat,modulo_modulo(nat,M,aa(nat,nat,suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_1185_div__times__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,Nb)),Nb)),M) ).

% div_times_less_eq_dividend
tff(fact_1186_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),divide_divide(nat,M,Nb))),M) ).

% times_div_less_eq_dividend
tff(fact_1187_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,A2,B2)),B2)),A2) ).

% div_mult_le
tff(fact_1188_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less_eq(int,K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$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_1189_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),M)
     => aa(int,$o,ord_less_eq(int,modulo_modulo(int,M,K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_1190_zdvd__antisym__nonneg,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),M)
     => ( aa(int,$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_1191_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,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_1192_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(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_1193_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( ! [E2: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),E2)
             => aa(A,$o,ord_less_eq(A,Xd),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2)) )
         => aa(A,$o,ord_less_eq(A,Xd),Y) ) ) ).

% field_le_epsilon
tff(fact_1194_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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_1195_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,$o,ord_less_eq(A,B2),C2)
           => aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1196_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,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(A,$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_1197_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$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_1198_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$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_1199_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,$o,ord_less_eq(A,B2),zero_zero(A))
           => aa(A,$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_1200_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1201_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1202_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,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1203_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1204_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,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),B2) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1205_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,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1206_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1207_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,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),B2) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1208_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,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1209_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,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,ord_less_eq(A,A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1210_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1211_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1212_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1213_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(A,$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),D)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1214_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xd: A,Y: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1215_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xd = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1216_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( aa(A,$o,ord_less_eq(A,Y),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xd)),Xd) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1217_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( aa(A,$o,ord_less_eq(A,Y),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y)),Xd) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1218_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),one_one(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
           => ( aa(A,$o,ord_less_eq(A,B2),one_one(A))
             => aa(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_1219_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,C2),one_one(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2) ) ) ) ).

% mult_left_le
tff(fact_1220_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> aa(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_1221_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xd: A,W: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,Xd),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less_eq(A,W),Z)
               => aa(A,$o,ord_less_eq(A,divide_divide(A,Xd,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_le
tff(fact_1222_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less(A,Xd),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less_eq(A,W),Z)
               => aa(A,$o,ord_less(A,divide_divide(A,Xd,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_less
tff(fact_1223_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less_eq(A,Xd),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less(A,W),Z)
               => aa(A,$o,ord_less(A,divide_divide(A,Xd,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1224_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,A2,C2)),divide_divide(A,B2,C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A2),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_1225_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,divide_divide(A,Xd,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1226_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),divide_divide(A,Xd,Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1227_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),divide_divide(A,Xd,Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1228_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,divide_divide(A,Xd,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1229_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1230_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => ( aa(A,$o,ord_less_eq(A,B2),A2)
           => aa(A,$o,ord_less(A,zero_zero(A)),divide_divide(A,A2,B2)) ) ) ) ).

% div_positive
tff(fact_1231_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,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_1232_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1233_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,A2),one_one(A))
           => aa(A,$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_1234_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,ord_less_eq(A,zero_zero(A)),A2)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1235_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$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,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(A,$o,ord_less_eq(A,A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_1236_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,ord_less_eq(A,zero_zero(A)),modulo_modulo(A,A2,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1237_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1238_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,ord_less(nat,K2),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,ord_less_eq(nat,I4),K2)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_1239_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb)) ) ).

% nat_one_le_power
tff(fact_1240_div__le__mono2,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less_eq(nat,M),Nb)
       => aa(nat,$o,ord_less_eq(nat,divide_divide(nat,K,Nb)),divide_divide(nat,K,M)) ) ) ).

% div_le_mono2
tff(fact_1241_div__greater__zero__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),divide_divide(nat,M,Nb))
    <=> ( aa(nat,$o,ord_less_eq(nat,Nb),M)
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_1242_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1243_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,ord_less_eq(nat,K),Nb) ) ) ).

% dvd_imp_le
tff(fact_1244_mod__le__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,ord_less_eq(nat,modulo_modulo(nat,M,Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_1245_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,one_one(int)),Z)
    <=> aa(int,$o,ord_less(int,zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_1246_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),Z)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_1247_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)
    <=> aa(int,$o,ord_less(int,W),Z) ) ).

% add1_zle_eq
tff(fact_1248_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,modulo_modulo(int,K,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1249_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => aa(int,$o,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1250_int__mod__lem,axiom,
    ! [Nb: int,B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B2)
          & aa(int,$o,ord_less(int,B2),Nb) )
      <=> ( modulo_modulo(int,B2,Nb) = B2 ) ) ) ).

% int_mod_lem
tff(fact_1251_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,ord_less_eq(int,zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,ord_less_eq(int,I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_1252_int__mod__eq,axiom,
    ! [B2: int,Nb: int,A2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B2)
     => ( aa(int,$o,ord_less(int,B2),Nb)
       => ( ( modulo_modulo(int,A2,Nb) = modulo_modulo(int,B2,Nb) )
         => ( modulo_modulo(int,A2,Nb) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_1253_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,A2,B2))
        & aa(int,$o,ord_less(int,modulo_modulo(int,A2,B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_1254_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,B2),zero_zero(int))
     => ( aa(int,$o,ord_less_eq(int,modulo_modulo(int,A2,B2)),zero_zero(int))
        & aa(int,$o,ord_less(int,B2),modulo_modulo(int,A2,B2)) ) ) ).

% neg_mod_conj
tff(fact_1255_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,ord_less_eq(int,A2),modulo_modulo(int,A2,Nb)) ) ) ).

% int_mod_ge
tff(fact_1256_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,ord_less_eq(nat,Nb),M)
       => ~ ! [S2: 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),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_1257_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,ord_less_eq(nat,M),Nb)
       => ~ ! [S2: 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),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_1258_nat__mod__eq__lemma,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xd,Nb) = modulo_modulo(nat,Y,Nb) )
     => ( aa(nat,$o,ord_less_eq(nat,Y),Xd)
       => ? [Q4: nat] : Xd = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1259_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),divide_divide(int,A2,B2))
      <=> ( aa(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_1260_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),divide_divide(int,A2,B2))
      <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1261_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,B2),zero_zero(int))
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),divide_divide(int,A2,B2))
      <=> aa(int,$o,ord_less_eq(int,A2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1262_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_eq(int,divide_divide(int,A2,B2)),A2) ) ) ).

% zdiv_le_dividend
tff(fact_1263_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)),divide_divide(int,I,K))
      <=> aa(int,$o,ord_less_eq(int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1264_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_eq(int,divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1265_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_eq(int,divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1266_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,L),K)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),L)
       => aa(int,$o,ord_less(int,zero_zero(int)),divide_divide(int,K,L)) ) ) ).

% div_positive_int
tff(fact_1267_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),divide_divide(int,K,L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),L) )
        | ( aa(int,$o,ord_less(int,K),zero_zero(int))
          & aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_1268_zdiv__mono2__neg,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( aa(int,$o,ord_less(int,A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
       => ( aa(int,$o,ord_less_eq(int,B6),B2)
         => aa(int,$o,ord_less_eq(int,divide_divide(int,A2,B6)),divide_divide(int,A2,B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1269_zdiv__mono1__neg,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,A2),A4)
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,divide_divide(int,A4,B2)),divide_divide(int,A2,B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1270_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( divide_divide(int,I,K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,ord_less_eq(int,zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,ord_less_eq(int,I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1271_zdiv__mono2,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
       => ( aa(int,$o,ord_less_eq(int,B6),B2)
         => aa(int,$o,ord_less_eq(int,divide_divide(int,A2,B2)),divide_divide(int,A2,B6)) ) ) ) ).

% zdiv_mono2
tff(fact_1272_zdiv__mono1,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,A2),A4)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,ord_less_eq(int,divide_divide(int,A2,B2)),divide_divide(int,A4,B2)) ) ) ).

% zdiv_mono1
tff(fact_1273_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,ord_less_eq(int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_1274_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),C2)
     => ( divide_divide(int,A2,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = divide_divide(int,divide_divide(int,A2,B2),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1275_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A2)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B2)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,A2,B2))
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),divide_divide(int,A2,B2)) ) ) ) ).

% nonneg_mod_div
tff(fact_1276_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,one_one(A)),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1277_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1278_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,one_one(A)),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1279_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1280_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,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B2) )
            & ( aa(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_1281_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,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),one_one(A)) )
            & ( aa(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_1282_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,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B2) )
            & ( aa(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_1283_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,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),one_one(A)) )
            & ( aa(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_1284_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] :
          ( ! [Z2: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),Z2)
             => ( aa(A,$o,ord_less(A,Z2),one_one(A))
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Xd)),Y) ) )
         => aa(A,$o,ord_less_eq(A,Xd),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1285_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xd: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),A2)
         => ( aa(A,$o,ord_less_eq(A,Y),A2)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),U)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1286_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,B2,A2)),one_one(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,B2),A2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1287_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),divide_divide(A,B2,A2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less_eq(A,A2),B2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1288_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,B2,C2)),A2)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(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,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_1289_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),divide_divide(A,B2,C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,ord_less_eq(A,A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1290_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B2),A2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,ord_less_eq(A,divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1291_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,$o,ord_less_eq(A,divide_divide(A,B2,C2)),A2)
          <=> aa(A,$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_1292_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,$o,ord_less_eq(A,A2),divide_divide(A,B2,C2))
          <=> aa(A,$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_1293_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,$o,ord_less_eq(A,divide_divide(A,B2,C2)),A2)
          <=> aa(A,$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_1294_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,$o,ord_less_eq(A,A2),divide_divide(A,B2,C2))
          <=> aa(A,$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_1295_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xd: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,Xd),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,ord_less_eq(A,divide_divide(A,Xd,Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1296_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xd: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xd)
           => aa(A,$o,ord_less_eq(A,Z),divide_divide(A,Xd,Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1297_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,ord_less_eq(A,divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1298_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,A2),one_one(A))
           => aa(A,$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_1299_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,ord_less_eq(A,zero_zero(A)),A2)
           => ( aa(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_1300_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,$o,ord_less_eq(A,zero_zero(A)),A2)
           => ( aa(A,$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_1301_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),A2)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => aa(A,$o,ord_less_eq(A,A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% self_le_power
tff(fact_1302_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xd: A,M: nat,Nb: nat] :
          ( ( Xd != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb))
          <=> ( aa(A,$o,dvd_dvd(A,Xd),one_one(A))
              | aa(nat,$o,ord_less_eq(nat,M),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_1303_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)) ) ).

% self_le_ge2_pow
tff(fact_1304_power2__nat__le__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_1305_power2__nat__le__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Nb)
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_1306_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,M: nat] :
      ( aa(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)))
       => ( divide_divide(nat,M,Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_1307_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,ord_less_eq(nat,M),divide_divide(nat,Nb,Q2))
      <=> aa(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_1308_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)),A2)
      <=> aa(nat,$o,ord_less_eq(nat,B2),divide_divide(nat,A2,C2)) ) ) ).

% td_gal
tff(fact_1309_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,ord_less_eq(nat,M),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_1310_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_1311_pos__mult__pos__ge,axiom,
    ! [Xd: int,Nb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Nb)
       => aa(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),Xd)) ) ) ).

% pos_mult_pos_ge
tff(fact_1312_q__pos__lemma,axiom,
    ! [B6: int,Q6: int,R4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4))
     => ( aa(int,$o,ord_less(int,R4),B6)
       => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
         => aa(int,$o,ord_less_eq(int,zero_zero(int)),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_1313_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B6: int,Q6: 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)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4))
       => ( aa(int,$o,ord_less(int,R4),B6)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
             => ( aa(int,$o,ord_less_eq(int,B6),B2)
               => aa(int,$o,ord_less_eq(int,Q2),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1314_incr__mult__lemma,axiom,
    ! [D: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X: int] :
            ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D)) )
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$o,P,X4)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1315_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B6: int,Q6: 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)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4) )
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4)),zero_zero(int))
       => ( aa(int,$o,ord_less(int,R2),B2)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R4)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
             => ( aa(int,$o,ord_less_eq(int,B6),B2)
               => aa(int,$o,ord_less_eq(int,Q6),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1316_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2))
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R4)
       => ( aa(int,$o,ord_less(int,R4),B2)
         => ( aa(int,$o,ord_less(int,R2),B2)
           => aa(int,$o,ord_less_eq(int,Q6),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1317_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2))
     => ( aa(int,$o,ord_less_eq(int,R2),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B2),R2)
         => ( aa(int,$o,ord_less(int,B2),R4)
           => aa(int,$o,ord_less_eq(int,Q2),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1318_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1319_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,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),Nb)),modulo_modulo(int,B2,Nb)) ) ) ).

% int_mod_ge'
tff(fact_1320_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,K,L))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),K) )
        | aa(int,$o,ord_less(int,zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_1321_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_1322_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_1323_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xd: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,ord_less(A,Xd),A2)
         => ( aa(A,$o,ord_less(A,Y),A2)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),U)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1324_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,ord_less_eq(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_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,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1325_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(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,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1326_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,$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_1327_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
             => ( Xd = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1328_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,Xd),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_1329_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,$o,ord_less_eq(A,zero_zero(A)),A2)
           => ( aa(nat,$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_1330_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( 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),modulo_modulo(A,divide_divide(A,A2,B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1331_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_1332_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,$o,ord_less_eq(A,A2),B2)
           => aa(A,$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_1333_pos__mod__sign2,axiom,
    ! [A2: int] : aa(int,$o,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,A2,aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_sign2
tff(fact_1334_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,divide_divide(nat,M,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q5: nat] :
            ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q5)),M)
            & aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q5)))
            & aa(nat,$o,P,Q5) ) ) ) ).

% split_div'
tff(fact_1335_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_1336_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))
      <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_1337_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa(real,$o,ord_less_eq(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)) ).

% two_realpow_ge_one
tff(fact_1338_split__zmod,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,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,J2: int] :
              ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J2)
                & aa(int,$o,ord_less(int,J2),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)),J2) ) )
             => aa(int,$o,P,J2) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I2: int,J2: int] :
              ( ( aa(int,$o,ord_less(int,K),J2)
                & aa(int,$o,ord_less_eq(int,J2),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)),J2) ) )
             => aa(int,$o,P,J2) ) ) ) ) ).

% split_zmod
tff(fact_1339_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: 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)),R2) )
     => ( aa(int,$o,ord_less_eq(int,R2),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B2),R2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1340_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: 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)),R2) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
       => ( aa(int,$o,ord_less(int,R2),B2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1341_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: 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)),R2) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
       => ( aa(int,$o,ord_less(int,R2),B2)
         => ( divide_divide(int,A2,B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1342_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: 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)),R2) )
     => ( aa(int,$o,ord_less_eq(int,R2),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B2),R2)
         => ( divide_divide(int,A2,B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1343_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,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,J2: int] :
              ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J2)
                & aa(int,$o,ord_less(int,J2),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)),J2) ) )
             => aa(int,$o,P,I2) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I2: int,J2: int] :
              ( ( aa(int,$o,ord_less(int,K),J2)
                & aa(int,$o,ord_less_eq(int,J2),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)),J2) ) )
             => aa(int,$o,P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_1344_mod__power__lem,axiom,
    ! [A2: int,Nb: nat,M: nat] :
      ( aa(int,$o,ord_less(int,one_one(int)),A2)
     => ( 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,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_1345_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),C2)
     => ( 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),modulo_modulo(int,divide_divide(int,A2,B2),C2))),modulo_modulo(int,A2,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1346_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,Xd),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_1347_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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_1348_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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))
        <=> ( ( Xd = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1349_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$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_1350_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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,ord_less_eq(A,zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1351_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,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
          <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_1352_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,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_1353_two__pow__div__gt__le,axiom,
    ! [V: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,V),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% two_pow_div_gt_le
tff(fact_1354_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D: real] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),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_1355_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B4)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(nat,$o,
            ord_less_eq(nat,
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A3,Nb)),
                $ite(modulo_modulo(nat,B4,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            divide_divide(nat,B4,Nb)) ) ) ).

% verit_le_mono_div
tff(fact_1356_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa(int,$o,ord_less(int,A3),B4)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
       => aa(int,$o,
            ord_less_eq(int,
              aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A3,Nb)),
                $ite(modulo_modulo(int,B4,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            divide_divide(int,B4,Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1357_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,divide_divide(int,Nb,K)),modulo_modulo(int,Nb,K))
      <=> ! [I2: int,J2: int] :
            ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J2)
              & aa(int,$o,ord_less(int,J2),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)),J2) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) ) ) ).

% split_pos_lemma
tff(fact_1358_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,divide_divide(int,Nb,K)),modulo_modulo(int,Nb,K))
      <=> ! [I2: int,J2: int] :
            ( ( aa(int,$o,ord_less(int,K),J2)
              & aa(int,$o,ord_less_eq(int,J2),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)),J2) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) ) ) ).

% split_neg_lemma
tff(fact_1359_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A] : aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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_1360_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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,ord_less_eq(A,zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1361_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$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),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_1362_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$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),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_1363_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,Xd: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),M)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
           => ( ( modulo_modulo(A,Xd,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = modulo_modulo(A,Xd,M) )
              | ( modulo_modulo(A,Xd,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),modulo_modulo(A,Xd,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1364_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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,$o,ord_less_eq(A,A2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1365_nat__div__eq__Suc__0__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( divide_divide(nat,Nb,M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(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_1366_power__2__mult__step__le,axiom,
    ! [N3: nat,Nb: nat,K4: nat,K: nat] :
      ( aa(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)),K4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),K))
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K4),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_1367_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A2)
     => ( 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))),modulo_modulo(int,B2,A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_1368_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A2)
     => ( 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)) = divide_divide(int,B2,A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1369_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,A2),zero_zero(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)) = 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_1370_eme1p,axiom,
    ! [Nb: int,D: 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))),D)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),D)
         => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb),D) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),modulo_modulo(int,Nb,D)) ) ) ) ) ).

% eme1p
tff(fact_1371_emep1,axiom,
    ! [Nb: int,D: 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))),D)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),D)
         => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)),D) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,Nb,D)),one_one(int)) ) ) ) ) ).

% emep1
tff(fact_1372_count__buildup,axiom,
    ! [Nb: nat] : aa(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_1373_two__realpow__ge__two,axiom,
    ! [Nb: nat] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(nat,real,semiring_1_of_nat(real),Nb))
     => aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)) ) ).

% two_realpow_ge_two
tff(fact_1374_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T)) ).

% cnt_non_neg
tff(fact_1375_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2))
     => aa(int,$o,ord_less_eq(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_1376_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,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)))),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_1377_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,A2),zero_zero(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))),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_1378_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))),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,ord_less_eq(nat,M),Nb)
              & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),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_1379_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : 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))),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_1380_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))),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_1381_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_1382_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_1383_verit__minus__simplify_I4_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),B2)) = B2 ) ).

% verit_minus_simplify(4)
tff(fact_1384_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_add_Oinverse__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(A,A,uminus_uminus(A),zero_zero(A)) = zero_zero(A) ) ) ).

% add.inverse_neutral
tff(fact_1393_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_1394_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_1395_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_1396_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_1397_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,ord_less_eq(A,A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_1398_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_1399_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_1400_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_1401_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_1402_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_1403_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_1404_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_1405_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_1406_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_1407_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_1408_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_1409_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_1410_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_1411_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_1412_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_1413_semiring__norm_I88_J,axiom,
    ! [M: num,Nb: num] : bit0(M) != aa(num,num,bit1,Nb) ).

% semiring_norm(88)
tff(fact_1414_semiring__norm_I89_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,bit1,M) != bit0(Nb) ).

% semiring_norm(89)
tff(fact_1415_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit1,Nb) ).

% semiring_norm(84)
tff(fact_1416_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_1417_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ).

% div_minus_minus
tff(fact_1418_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,Xd),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,dvd_dvd(A,Xd),Y) ) ) ).

% dvd_minus_iff
tff(fact_1419_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,uminus_uminus(A),Xd)),Y)
        <=> aa(A,$o,dvd_dvd(A,Xd),Y) ) ) ).

% minus_dvd_iff
tff(fact_1420_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_1421_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_1422_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% minus_mod_self2
tff(fact_1423_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1424_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_1425_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)) ) ).

% mod_minus_minus
tff(fact_1426_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_1427_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(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_1428_semiring__norm_I73_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,ord_less_eq(num,aa(num,num,bit1,M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,ord_less_eq(num,M),Nb) ) ).

% semiring_norm(73)
tff(fact_1429_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_1430_count__buildup_H,axiom,
    ! [Nb: nat] : aa(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_1431_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1432_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1433_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_1434_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_1435_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_1436_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_1437_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_1438_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_1439_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_1440_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_1441_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_1442_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_1443_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_1444_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_1445_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$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_1446_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$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_1447_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_1448_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_1449_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_1450_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_1451_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_1452_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_1453_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_1454_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_1455_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_1456_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_1457_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_1458_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xd: A] : divide_divide(A,Xd,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Xd) ) ).

% divide_minus1
tff(fact_1459_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: 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_1460_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_1461_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_1462_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_1463_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_1464_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_1465_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ) ).

% of_nat_le_iff
tff(fact_1466_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)
           => ( divide_divide(A,aa(A,A,minus_minus(A,A2),B2),C2) = aa(A,A,minus_minus(A,divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ) ).

% div_diff
tff(fact_1467_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_1468_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,B2),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_mod_self1
tff(fact_1469_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_1470_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_1471_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_1472_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_1473_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_1474_diff__is__0__eq_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1475_diff__is__0__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Nb) = zero_zero(nat) )
    <=> aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% diff_is_0_eq
tff(fact_1476_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_1477_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1478_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1479_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1480_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xd: nat,B2: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Xd) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) )
        <=> ( Xd = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1481_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,Xd: 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),Xd) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = Xd ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1482_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_1483_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_1484_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_1485_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_1486_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_1487_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_1488_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_1489_semiring__norm_I72_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,ord_less_eq(num,bit0(M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,ord_less_eq(num,M),Nb) ) ).

% semiring_norm(72)
tff(fact_1490_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_1491_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less_eq(num,aa(num,num,bit1,M)),one2) ).

% semiring_norm(70)
tff(fact_1492_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa(num,$o,ord_less(num,one2),aa(num,num,bit1,Nb)) ).

% semiring_norm(77)
tff(fact_1493_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_1494_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_1495_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_1496_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_1497_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_1498_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_1499_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_1500_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_1501_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_1502_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_1503_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_1504_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_1505_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_1506_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1507_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1508_Suc__diff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( aa(nat,$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_1509_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : divide_divide(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V)),aa(num,int,numeral_numeral(int),bit0(W))) = divide_divide(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_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_1518_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_1519_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_1520_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_1521_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_1522_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_1523_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_1524_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,W),aa(int,int,minus_minus(int,Z),one_one(int)))
    <=> aa(int,$o,ord_less(int,W),Z) ) ).

% zle_diff1_eq
tff(fact_1525_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,ord_less_eq(num,Nb),M) ) ) ).

% neg_numeral_le_iff
tff(fact_1526_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_1527_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_1528_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,ord_less_eq(num,M),Nb) ) ).

% semiring_norm(79)
tff(fact_1529_semiring__norm_I74_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(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_1530_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_1531_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_1532_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = 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_1533_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: 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_1534_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(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_1535_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(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_1536_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),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_1537_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,ord_less(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_1538_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_1539_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_1540_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_1541_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,Xd: 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),Xd)),Nb) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xd)),Nb) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1542_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xd: num,Nb: nat,Y: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xd)),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),Xd)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1543_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: nat,B2: nat,W: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Xd)),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,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1544_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xd: nat] :
          ( aa(A,$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),Xd))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xd) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1545_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: nat,B2: nat,W: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),Xd)),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_eq(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1546_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xd: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xd))
        <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xd) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1547_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_1548_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_1549_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,M: nat] :
      ( aa(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,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),Nb)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_1550_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less(int,A2),zero_zero(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_1551_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_1552_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_1553_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_1554_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(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_1555_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_1556_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(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_1557_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(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_1558_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: nat,Nb: nat] :
          ( aa(A,$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),Xd)),Nb))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Xd)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1559_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_1560_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_1561_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_1562_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V)) = 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_1563_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Nb: nat] : divide_divide(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = 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_1564_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V)) = 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_1565_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Nb: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = 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_1566_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_1567_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_1568_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_1569_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_1570_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_1571_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_1572_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: nat,I: num,Nb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Xd)),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,Xd),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_1573_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xd: nat] :
          ( aa(A,$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),Xd))
        <=> 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)),Xd) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1574_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: nat,I: num,Nb: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),Xd)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$o,ord_less_eq(nat,Xd),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_1575_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xd: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xd))
        <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Nb)),Xd) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1576_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_1577_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_1578_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),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_1579_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_1580_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_1581_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_1582_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_1583_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_1584_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,T)) ).

% cnt_cnt_eq
tff(fact_1585_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_1586_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$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_1587_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less_eq(nat,Nb),Nb) ).

% le_refl
tff(fact_1588_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,J),K)
       => aa(nat,$o,ord_less_eq(nat,I),K) ) ) ).

% le_trans
tff(fact_1589_eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = Nb )
     => aa(nat,$o,ord_less_eq(nat,M),Nb) ) ).

% eq_imp_le
tff(fact_1590_le__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,Nb),M)
       => ( M = Nb ) ) ) ).

% le_antisym
tff(fact_1591_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$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_1592_le__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),Nb)
       => ( aa(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,ord_less_eq(nat,M),Nb) ) ) ) ).

% le_diff_iff
tff(fact_1593_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$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_1594_diff__le__mono,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => aa(nat,$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_1595_diff__le__self,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),Nb)),M) ).

% diff_le_self
tff(fact_1596_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A2),C2)
     => ( aa(nat,$o,ord_less_eq(nat,B2),C2)
       => ( aa(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,ord_less_eq(nat,B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_1597_diff__le__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => aa(nat,$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_1598_nat__le__linear,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
      | aa(nat,$o,ord_less_eq(nat,Nb),M) ) ).

% nat_le_linear
tff(fact_1599_complete__real,axiom,
    ! [S3: set(real)] :
      ( ? [X4: real] : member(real,X4,S3)
     => ( ? [Z3: real] :
          ! [X: real] :
            ( member(real,X,S3)
           => aa(real,$o,ord_less_eq(real,X),Z3) )
       => ? [Y2: real] :
            ( ! [X4: real] :
                ( member(real,X4,S3)
               => aa(real,$o,ord_less_eq(real,X4),Y2) )
            & ! [Z3: real] :
                ( ! [X: real] :
                    ( member(real,X,S3)
                   => aa(real,$o,ord_less_eq(real,X),Z3) )
               => aa(real,$o,ord_less_eq(real,Y2),Z3) ) ) ) ) ).

% complete_real
tff(fact_1600_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,ord_less_eq(nat,Y2),B2) )
       => ? [X: nat] :
            ( aa(nat,$o,P,X)
            & ! [Y3: nat] :
                ( aa(nat,$o,P,Y3)
               => aa(nat,$o,ord_less_eq(nat,Y3),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_1601_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_1602_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_1603_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_1604_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_1605_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B4: A,K: A,B2: A,A2: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,minus_minus(A,A2),B4) = 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_1606_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_1607_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_1608_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_1609_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% equation_minus_iff
tff(fact_1610_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D) )
         => ( ( A2 = B2 )
          <=> ( C2 = D ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1611_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_1612_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_1613_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_1614_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xd)),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),Xd),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_1615_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B2),modulo_modulo(int,A2,B2))) ).

% zmod_zminus1_eq_if
tff(fact_1616_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,modulo_modulo(int,A2,B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_1617_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,ord_less_eq(nat,I),J)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_1618_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xd: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xd)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xd)) ) ).

% mult_of_nat_commute
tff(fact_1619_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_1620_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D) )
         => ( aa(A,$o,ord_less_eq(A,A2),B2)
          <=> aa(A,$o,ord_less_eq(A,C2),D) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1621_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => aa(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_1622_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B2),A2)
         => aa(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_1623_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,D),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D)) ) ) ) ).

% diff_mono
tff(fact_1624_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,D),C2)
           => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D)) ) ) ) ).

% diff_strict_mono
tff(fact_1625_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D) )
         => ( aa(A,$o,ord_less(A,A2),B2)
          <=> aa(A,$o,ord_less(A,C2),D) ) ) ) ).

% diff_eq_diff_less
tff(fact_1626_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_1627_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_1628_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => aa(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_1629_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_le_iff
tff(fact_1630_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,ord_less_eq(A,B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_minus_iff
tff(fact_1631_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D: 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),D)) = 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),D)) ) ).

% add_diff_add
tff(fact_1632_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_1633_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_1634_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_1635_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_1636_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_1637_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_1638_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_1639_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_1640_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_1641_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D3: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D3))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D3))) )
           => ! [X4: A,K5: A] :
                ( ( aa(A,$o,P,X4)
                  & aa(A,$o,Q,X4) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3)))
                  & aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1642_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D3: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D3))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D3))) )
           => ! [X4: A,K5: A] :
                ( ( aa(A,$o,P,X4)
                  | aa(A,$o,Q,X4) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3)))
                  | aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_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_1653_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_1654_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_1655_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_1656_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_1657_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_1658_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,aa(A,A,minus_minus(A,A2),B2),C2) = aa(A,A,minus_minus(A,divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ).

% diff_divide_distrib
tff(fact_1659_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% div_minus_right
tff(fact_1660_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_divide_left
tff(fact_1661_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ).

% minus_divide_divide
tff(fact_1662_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_1663_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : bit0(X2) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_1664_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_1665_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_1666_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xd: A,Y: A,Z: A] :
          ( aa(A,$o,dvd_dvd(A,Xd),Y)
         => ( aa(A,$o,dvd_dvd(A,Xd),Z)
           => aa(A,$o,dvd_dvd(A,Xd),aa(A,A,minus_minus(A,Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_1667_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,aa(nat,nat,minus_minus(nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_1668_Suc__diff__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(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_1669_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_right_eq
tff(fact_1670_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_left_eq
tff(fact_1671_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A4,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A4),B6),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1672_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_eq
tff(fact_1673_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_1674_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1675_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2)) ) ).

% mod_minus_right
tff(fact_1676_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A4: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A4,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A4),B2) ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_1677_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% mod_minus_eq
tff(fact_1678_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_1679_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_1680_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_1681_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),B2)
     => ( aa(nat,$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_1682_less__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$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_1683_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_1684_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_1685_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_1686_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_1687_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_1688_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( ( aa(nat,nat,minus_minus(nat,J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1689_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1690_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1691_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1692_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,ord_less_eq(nat,J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_1693_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_1694_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_1695_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_1696_Word_Omod__minus__cong,axiom,
    ! [B2: int,B6: int,Xd: int,X6: int,Y: int,Y4: int,Z4: int] :
      ( ( B2 = B6 )
     => ( ( modulo_modulo(int,Xd,B6) = modulo_modulo(int,X6,B6) )
       => ( ( modulo_modulo(int,Y,B6) = modulo_modulo(int,Y4,B6) )
         => ( ( aa(int,int,minus_minus(int,X6),Y4) = Z4 )
           => ( modulo_modulo(int,aa(int,int,minus_minus(int,Xd),Y),B2) = modulo_modulo(int,Z4,B6) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_1697_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_1698_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,$o,ord_less_eq(nat,Nb),M)
         => aa(nat,$o,dvd_dvd(nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_1699_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,$o,ord_less_eq(nat,Nb),M)
         => aa(nat,$o,dvd_dvd(nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_1700_less__eq__dvd__minus,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1701_le__mod__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Nb),M)
     => ( modulo_modulo(nat,M,Nb) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb) ) ) ).

% le_mod_geq
tff(fact_1702_uminus__dvd__conv_I1_J,axiom,
    ! [D: int,T: int] :
      ( aa(int,$o,dvd_dvd(int,D),T)
    <=> aa(int,$o,dvd_dvd(int,aa(int,int,uminus_uminus(int),D)),T) ) ).

% uminus_dvd_conv(1)
tff(fact_1703_uminus__dvd__conv_I2_J,axiom,
    ! [D: int,T: int] :
      ( aa(int,$o,dvd_dvd(int,D),T)
    <=> aa(int,$o,dvd_dvd(int,D),aa(int,int,uminus_uminus(int),T)) ) ).

% uminus_dvd_conv(2)
tff(fact_1704_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,Xd,Z))),Y) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1705_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,divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),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_1706_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),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),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_1707_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$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_1708_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),divide_divide(nat,M,Nb)) = 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),modulo_modulo(nat,M,Nb))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% field_char_0_class.of_nat_div
tff(fact_1709_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_0_le_iff
tff(fact_1710_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_1711_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_1712_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_1713_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_1714_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Nb: nat] : 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))) = divide_divide(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_1715_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),L)
     => ( 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))),modulo_modulo(int,aa(int,int,minus_minus(int,K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_1716_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( 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_1717_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_1718_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A2),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_1719_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_1720_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => aa(A,$o,ord_less_eq(A,I),aa(A,A,minus_minus(A,Nb),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1721_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A,J: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => ( aa(A,$o,ord_less_eq(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
             => ( aa(A,$o,ord_less_eq(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,Nb),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1722_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A2),B2)),C2)
        <=> aa(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_1723_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(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_1724_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$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_1725_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_1726_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),aa(A,A,minus_minus(A,B2),A2))
          <=> aa(A,$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_1727_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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_1728_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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_1729_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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_1730_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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_1731_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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_1732_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$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_1733_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,ord_less_eq(A,A2),B2)
         => ( aa(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_1734_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_1735_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_1736_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_1737_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( divide_divide(int,A2,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_1738_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( divide_divide(int,aa(int,int,uminus_uminus(int),A2),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_1739_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_1740_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : ~ aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_le_neg_numeral
tff(fact_1741_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_le_numeral
tff(fact_1742_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xd: A,Y: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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),Xd),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,Xd),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_1743_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xd: A,Y: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Xd)),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),Xd),Y)),aa(A,A,minus_minus(A,Xd),Y)) ) ).

% square_diff_square_factored
tff(fact_1744_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E)),D) ) ) ) ).

% eq_add_iff2
tff(fact_1745_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E)),C2) = D ) ) ) ).

% eq_add_iff1
tff(fact_1746_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_1747_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_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_1755_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_1756_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),divide_divide(nat,M,Nb)) = 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_1757_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_1758_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_1759_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_1760_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_1761_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,Xd: 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),Xd)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xd),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_1762_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1763_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1764_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xd: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Xd) = one_one(A) )
        <=> ( ( Xd = one_one(A) )
            | ( Xd = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1765_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X22: num] : Y != bit0(X22)
       => ~ ! [X33: num] : Y != aa(num,num,bit1,X33) ) ) ).

% num.exhaust
tff(fact_1766_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_1767_nat__less__real__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
    <=> aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Nb)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_1768_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_div_neg
tff(fact_1769_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_neg_div
tff(fact_1770_nat__le__real__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(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_1771_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
        <=> aa(A,$o,dvd_dvd(A,C2),aa(A,A,minus_minus(A,A2),B2)) ) ) ).

% mod_eq_dvd_iff
tff(fact_1772_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),modulo_modulo(A,A2,B2))) ) ).

% dvd_minus_mod
tff(fact_1773_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,Nb)) = 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_1774_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_1775_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_1776_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_1777_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_1778_real__of__nat__div4,axiom,
    ! [Nb: nat,Xd: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,Xd))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),Xd))) ).

% real_of_nat_div4
tff(fact_1779_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_1780_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_1781_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_1782_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_1783_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,J),K)),I)
      <=> aa(nat,$o,ord_less(nat,J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_1784_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less_eq(int,I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$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_1785_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_1786_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),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_1787_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),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_1788_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_1789_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_1790_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),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_1791_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),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_1792_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_1793_nat__minus__mod,axiom,
    ! [Nb: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,minus_minus(nat,Nb),modulo_modulo(nat,Nb,M)),M) = zero_zero(nat) ).

% nat_minus_mod
tff(fact_1794_mod__nat__sub,axiom,
    ! [Xd: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),Z)
     => ( modulo_modulo(nat,aa(nat,nat,minus_minus(nat,Xd),Y),Z) = aa(nat,nat,minus_minus(nat,Xd),Y) ) ) ).

% mod_nat_sub
tff(fact_1795_mod__geq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),Nb)
     => ( modulo_modulo(nat,M,Nb) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb) ) ) ).

% mod_geq
tff(fact_1796_mod__if,axiom,
    ! [M: nat,Nb: nat] :
      modulo_modulo(nat,M,Nb) = $ite(aa(nat,$o,ord_less(nat,M),Nb),M,modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ).

% mod_if
tff(fact_1797_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_1798_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_1799_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_1800_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,K,aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_1801_nat__minus__mod__plus__right,axiom,
    ! [Nb: nat,Xd: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Xd)),modulo_modulo(nat,Nb,M)),M) = modulo_modulo(nat,Xd,M) ).

% nat_minus_mod_plus_right
tff(fact_1802_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X: nat,Y2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D2),A2)
      & aa(nat,$o,dvd_dvd(nat,D2),B2)
      & ( ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)) = D2 )
        | ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_1803_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,M: nat,Q2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Nb),M)
     => ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,Nb,Q2) )
      <=> aa(nat,$o,dvd_dvd(nat,Q2),aa(nat,nat,minus_minus(nat,M),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1804_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_1805_small__powers__of__2,axiom,
    ! [Xd: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Xd)
     => aa(nat,$o,ord_less(nat,Xd),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,Xd),one_one(nat)))) ) ).

% small_powers_of_2
tff(fact_1806_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D))
        <=> aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E)),D)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1807_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E)),C2)),D) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1808_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$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_1809_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$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_1810_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D))
        <=> aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E)),D)) ) ) ).

% less_add_iff2
tff(fact_1811_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D))
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E)),C2)),D) ) ) ).

% less_add_iff1
tff(fact_1812_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_1813_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_1814_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_1815_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_1816_m1mod2k,axiom,
    ! [Nb: nat] : 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_1817_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Xd)),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),Xd),one_one(A))),aa(A,A,minus_minus(A,Xd),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1818_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,divide_divide(A,Xd,Z)),Y) = divide_divide(A,aa(A,A,minus_minus(A,Xd),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1819_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,Xd),divide_divide(A,Y,Z)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Z)),Y),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1820_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xd: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,divide_divide(A,Xd,Y)),divide_divide(A,W,Z)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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_1821_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),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_1822_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_1823_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_1824_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_1825_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_1826_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_1827_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_1828_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_1829_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_1830_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_1831_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_1832_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_1833_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_1834_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_1835_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_1836_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_1837_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_1838_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),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_1839_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),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_1840_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(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_1841_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(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_1842_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D: A,D3: A,T: A] :
          ( aa(A,$o,dvd_dvd(A,D),D3)
         => ! [X4: A,K5: A] :
              ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T))
            <=> ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))),T)) ) ) ) ).

% inf_period(4)
tff(fact_1843_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D: A,D3: A,T: A] :
          ( aa(A,$o,dvd_dvd(A,D),D3)
         => ! [X4: A,K5: A] :
              ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T))
            <=> aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))),T)) ) ) ) ).

% inf_period(3)
tff(fact_1844_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_1845_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xd)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ).

% power_minus_Bit0
tff(fact_1846_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),divide_divide(A,A2,B2)) = aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) ) ).

% zmde
tff(fact_1847_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),divide_divide(A,A2,B2))) = modulo_modulo(A,A2,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1848_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1849_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1850_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),divide_divide(A,A2,B2)),B2)) = modulo_modulo(A,A2,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1851_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_1852_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_1853_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Nb: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$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)) = 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_1854_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q2))) != 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_1855_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q2))) != 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_1856_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1857_reals__Archimedean3,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ! [Y3: real] :
        ? [N: nat] : aa(real,$o,ord_less(real,Y3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xd)) ) ).

% reals_Archimedean3
tff(fact_1858_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_1859_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,K),aa(nat,nat,suc,zero_zero(nat)))),M) ) ) ).

% nz_le_conv_less
tff(fact_1860_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(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_1861_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_1862_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_1863_real__of__nat__div,axiom,
    ! [D: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D),Nb)
     => ( aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,D)) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),D)) ) ) ).

% real_of_nat_div
tff(fact_1864_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),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_1865_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),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_1866_real__of__nat__div__aux,axiom,
    ! [Xd: nat,D: nat] : divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Xd),aa(nat,real,semiring_1_of_nat(real),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Xd,D))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,Xd,D)),aa(nat,real,semiring_1_of_nat(real),D))) ).

% real_of_nat_div_aux
tff(fact_1867_minusinfinity,axiom,
    ! [D: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P1,X)
          <=> aa(int,$o,P1,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
       => ( ? [Z3: int] :
            ! [X: int] :
              ( aa(int,$o,ord_less(int,X),Z3)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P1,X) ) )
         => ( ? [X_12: int] : aa(int,$o,P1,X_12)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_1868_plusinfinity,axiom,
    ! [D: int,P2: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P2,X)
          <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
       => ( ? [Z3: int] :
            ! [X: int] :
              ( aa(int,$o,ord_less(int,Z3),X)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P2,X) ) )
         => ( ? [X_12: int] : aa(int,$o,P2,X_12)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_1869_less__1__helper,axiom,
    ! [Nb: int,M: int] :
      ( aa(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_1870_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I3: int] :
            ( aa(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,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_1871_int__mod__le_H,axiom,
    ! [B2: int,Nb: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,B2),Nb))
     => aa(int,$o,ord_less_eq(int,modulo_modulo(int,B2,Nb)),aa(int,int,minus_minus(int,B2),Nb)) ) ).

% int_mod_le'
tff(fact_1872_mod__nat__add,axiom,
    ! [Xd: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),Z)
     => ( aa(nat,$o,ord_less(nat,Y),Z)
       => ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xd),Y),Z) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xd),Y)),Z),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xd),Y),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xd),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_1873_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Q2),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M))
       => ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Nb),Q2))
        <=> aa(nat,$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),R2),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_1874_diff__mod__le,axiom,
    ! [A2: nat,D: nat,B2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),D)
     => ( aa(nat,$o,dvd_dvd(nat,B2),D)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,A2),modulo_modulo(nat,A2,B2))),aa(nat,nat,minus_minus(nat,D),B2)) ) ) ).

% diff_mod_le
tff(fact_1875_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,R2),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,R2),M)
       => ( aa(nat,$o,dvd_dvd(nat,Nb),aa(nat,nat,minus_minus(nat,M),R2))
         => ( modulo_modulo(nat,M,Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_1876_modulo__nat__def,axiom,
    ! [M: nat,Nb: nat] : modulo_modulo(nat,M,Nb) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,Nb)),Nb)) ).

% modulo_nat_def
tff(fact_1877_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),divide_divide(int,A2,B2)),B2) = aa(int,int,minus_minus(int,A2),modulo_modulo(int,A2,B2)) ).

% mod_div_equality_div_eq
tff(fact_1878_m1mod22k,axiom,
    ! [Nb: nat] : 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_1879_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Nb),M)
         => ( ( Nb != zero_zero(nat) )
           => aa(A,$o,ord_less_eq(A,divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),M))),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_1880_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Nb: nat] : 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)),modulo_modulo(A,divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_1881_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xd: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,ord_less_eq(A,divide_divide(A,Xd,Y)),divide_divide(A,W,Z))
            <=> aa(A,$o,ord_less_eq(A,divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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_1882_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xd: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,ord_less(A,divide_divide(A,Xd,Y)),divide_divide(A,W,Z))
            <=> aa(A,$o,ord_less(A,divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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_1883_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,$o,ord_less(A,aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1884_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,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1885_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,$o,ord_less(A,aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1886_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,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1887_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),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_1888_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),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_1889_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)) = 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_1890_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( 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_1891_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xd: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,Xd),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),Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_commute
tff(fact_1892_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xd,Z))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1893_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),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),B2,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_1894_odd__mod__4__div__2,axiom,
    ! [Nb: 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))),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_1895_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_1896_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Xd: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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))) )
        <=> ( ( Xd = Y )
            | ( Xd = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_1897_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : 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_1898_real__archimedian__rdiv__eq__0,axiom,
    ! [Xd: real,C2: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),C2)
       => ( ! [M2: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M2)
             => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),Xd)),C2) )
         => ( Xd = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1899_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] : 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_1900_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_1901_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( ( A2 != zero_zero(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,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)),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_1902_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_1903_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_1904_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_1905_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),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_1906_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_1907_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_1908_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_1909_div__if,axiom,
    ! [M: nat,Nb: 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,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ) ).

% div_if
tff(fact_1910_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)
       => ( divide_divide(nat,M,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ) ) ) ).

% div_geq
tff(fact_1911_le__div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,Nb),M)
       => ( divide_divide(nat,M,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ) ) ) ).

% le_div_geq
tff(fact_1912_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_1913_power__sub,axiom,
    ! [Nb: nat,M: nat,A2: nat] :
      ( aa(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)) = 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_1914_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_1915_decr__mult__lemma,axiom,
    ! [D: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X: int] :
            ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X),D)) )
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$o,P,X4)
             => aa(int,$o,P,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1916_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$o,ord_less_eq(int,L),K)
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,minus_minus(int,K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1917_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa(int,$o,ord_less_eq(int,A3),B4)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,uminus_uminus(int),Nb))
       => aa(int,$o,ord_less_eq(int,divide_divide(int,B4,Nb)),divide_divide(int,A3,Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1918_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( 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_1919_int__div__sub__1,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,ord_less_eq(int,one_one(int)),M)
     => ( 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,divide_divide(int,Nb,M)),one_one(int)),divide_divide(int,Nb,M)) ) ) ).

% int_div_sub_1
tff(fact_1920_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_1921_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_1922_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R2: A,S: A] :
          ( aa(A,$o,ord_less_eq(A,U),V)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),R2)
           => ( aa(A,$o,ord_less_eq(A,R2),S)
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,minus_minus(A,V),U)),S))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1923_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),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_1924_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,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1925_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,$o,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1926_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,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1927_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,$o,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1928_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(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,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_1929_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,ord_less_eq(A,A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_1930_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))),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_1931_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(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_1932_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_1933_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_1934_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( 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_1935_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( 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_1936_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_1937_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P3: A,M: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1938_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_1939_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Nb: nat] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),Nb) = 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_1940_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(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)) = 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_1941_nat__le__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa(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,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),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_1942_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),Nb) = 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_1943_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_1944_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_1945_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_1946_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_1947_mod__add__if__z,axiom,
    ! [Xd: int,Z: int,Y: int] :
      ( aa(int,$o,ord_less(int,Xd),Z)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
           => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
             => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),Y),Z) = $ite(aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),Y)),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),Y),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_1948_mod__sub__if__z,axiom,
    ! [Xd: int,Z: int,Y: int] :
      ( aa(int,$o,ord_less(int,Xd),Z)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
           => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
             => ( modulo_modulo(int,aa(int,int,minus_minus(int,Xd),Y),Z) = $ite(aa(int,$o,ord_less_eq(int,Y),Xd),aa(int,int,minus_minus(int,Xd),Y),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,Xd),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_1949_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B2,C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(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,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_1950_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,ord_less_eq(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_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,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,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_1951_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),Xd)
         => ( aa(A,$o,ord_less_eq(A,Xd),one_one(A))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_1952_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_1953_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_1954_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ( 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),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_1955_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K)
     => aa(int,$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_1956_less__two__pow__divI,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),M)))
     => ( aa(nat,$o,ord_less_eq(nat,M),Nb)
       => aa(nat,$o,ord_less(nat,Xd),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_1957_less__two__pow__divD,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => ( aa(nat,$o,ord_less_eq(nat,M),Nb)
        & aa(nat,$o,ord_less(nat,Xd),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_1958_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_1959_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,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_1960_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),K)
        & aa(int,$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_1961_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),K)
     => ( aa(int,$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_1962_power__mod__div,axiom,
    ! [Xd: nat,Nb: nat,M: nat] : divide_divide(nat,modulo_modulo(nat,Xd,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)) = modulo_modulo(nat,divide_divide(nat,Xd,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_1963_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : 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_1964_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$o,ord_less_eq(int,L),K)
       => ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,aa(int,int,minus_minus(int,K),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1965_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( divide_divide(int,K,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1966_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat,M: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xd),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,$o,ord_less_eq(nat,M),Nb)
            & aa(word(A),$o,ord_less(word(A),Xd),aa(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_1967_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xd: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,Xd),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),Xd),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))),Xd)),Y)) ) ).

% power2_diff
tff(fact_1968_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))),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,ord_less_eq(nat,M),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_1969_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_1970_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xd))),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)),Xd)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_1971_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$o,P,K2)
             => ( ( K2 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
         => ( ! [K2: int] :
                ( aa(int,$o,P,K2)
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_1972_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)
       => ( 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_1973_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => ( aa(A,$o,ord_less_eq(A,B2),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,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) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1974_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))),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,ord_less_eq(nat,M),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_1975_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($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,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_1976_even__mod__4__div__2,axiom,
    ! [Nb: 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))),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_1977_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,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_1978_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),A2)
     => aa(int,$o,ord_less_eq(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_1979_space__cnt,axiom,
    ! [T: vEBT_VEBT] : aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T))),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,T))) ).

% space_cnt
tff(fact_1980_t__build__cnt,axiom,
    ! [Nb: nat] : aa(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_1981_t__buildup__cnt,axiom,
    ! [Nb: nat] : aa(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_1982_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,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_1983_linear__plus__1__le__power,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xd)),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),Xd),one_one(real))),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_1984_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( 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_1985_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),E)
         => ~ ! [N: nat] : ~ aa(A,$o,ord_less(A,divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E) ) ) ).

% nat_approx_posE
tff(fact_1986_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),Xd)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,ord_less(A,Y),Xd) ) ) ).

% compl_less_compl_iff
tff(fact_1987_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_1988_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_1989_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_1990_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_1991_negative__eq__positive,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( Nb = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_1992_real__add__minus__iff,axiom,
    ! [Xd: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( Xd = A2 ) ) ).

% real_add_minus_iff
tff(fact_1993_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$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_1994_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$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_1995_word__n1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$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_1996_max__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : aa(word(A),$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_1997_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xd)
        <=> ( Xd = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_minus_one_le
tff(fact_1998_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$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_1999_int__dvd__int__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(int,$o,dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,dvd_dvd(nat,M),Nb) ) ).

% int_dvd_int_iff
tff(fact_2000_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_2001_div__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : divide_divide(word(A),W,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa($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_2002_word__le__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A)))) ) ) ) ).

% word_le_sub1_numberof
tff(fact_2003_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_2004_mod__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(word(A),one_one(word(A)),W) = aa(word(A),word(A),minus_minus(word(A),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))))) ) ).

% mod_word_one
tff(fact_2005_mod__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(word(A),W,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($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_2006_minus__real__def,axiom,
    ! [Xd: real,Y: real] : aa(real,real,minus_minus(real,Xd),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_2007_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Xd),Z))
        <=> aa(word(A),$o,ord_less(word(A),Xd),Z) ) ) ).

% sub_wrap_lt
tff(fact_2008_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Xd),Y))
        <=> aa(word(A),$o,ord_less(word(A),Xd),Y) ) ) ).

% word_sub_less_iff
tff(fact_2009_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),D: word(A),B2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),C2)
         => ( aa(word(A),$o,ord_less(word(A),D),B2)
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),A2),B2)),A2)
             => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),C2),D)),C2)
               => aa(word(A),$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),D)) ) ) ) ) ) ).

% word_less_minus_mono
tff(fact_2010_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,ord_less_eq(real,A2),B2)
     => ( ! [A5: real,B3: real,C4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A5),B3)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B3),C4)
             => ( aa(real,$o,ord_less_eq(real,A5),B3)
               => ( aa(real,$o,ord_less_eq(real,B3),C4)
                 => aa(real,$o,aa(real,fun(real,$o),P,A5),C4) ) ) ) )
       => ( ! [X: real] :
              ( aa(real,$o,ord_less_eq(real,A2),X)
             => ( aa(real,$o,ord_less_eq(real,X),B2)
               => ? [D5: real] :
                    ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                    & ! [A5: real,B3: real] :
                        ( ( aa(real,$o,ord_less_eq(real,A5),X)
                          & aa(real,$o,ord_less_eq(real,X),B3)
                          & aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,B3),A5)),D5) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A5),B3) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_2011_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_2012_Abs__fnat__hom__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = aa(nat,word(A),semiring_1_of_nat(word(A)),zero_zero(nat)) ) ) ).

% Abs_fnat_hom_0
tff(fact_2013_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_2014_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% int_cases
tff(fact_2015_int__of__nat__induct,axiom,
    ! [P: fun(int,$o),Z: int] :
      ( ! [N: nat] : aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),N))
     => ( ! [N: nat] : aa(int,$o,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))
       => aa(int,$o,P,Z) ) ) ).

% int_of_nat_induct
tff(fact_2016_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_2017_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A2),B2)
    <=> aa(int,$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_2018_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_2019_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ? [N: nat] : K = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% zero_le_imp_eq_int
tff(fact_2020_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ~ ! [N: nat] : K != aa(nat,int,semiring_1_of_nat(int),N) ) ).

% nonneg_int_cases
tff(fact_2021_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_2022_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_2023_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_2024_not__int__zless__negative,axiom,
    ! [Nb: nat,M: nat] : ~ aa(int,$o,ord_less(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))) ).

% not_int_zless_negative
tff(fact_2025_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_2026_real__minus__mult__self__le,axiom,
    ! [U: real,Xd: real] : aa(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),Xd),Xd)) ).

% real_minus_mult_self_le
tff(fact_2027_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,A2,B2)) = 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_2028_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
         => ( ( Xd != Y )
           => aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ) ).

% le_step_down_word_2
tff(fact_2029_le__step__down__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),Nb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),Nb)
         => ( ( I != Nb )
           => aa(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_2030_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_2031_word__less__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),Y)
         => ( ( Xd = aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))) )
            | aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ) ).

% word_less_cases
tff(fact_2032_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Nb: word(A),M: word(A)] :
          ( aa(word(A),$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_2033_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),aa(word(A),word(A),minus_minus(word(A),Z),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Z)
           => aa(word(A),$o,ord_less(word(A),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_2034_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),aa(word(A),word(A),minus_minus(word(A),Z),Xd)) ) ) ) ).

% word_less_minus_mono_left
tff(fact_2035_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$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_2036_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,A2,B2)) = modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_2037_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_2038_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_2039_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_2040_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_2041_max__word__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : ~ aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xd) ) ).

% max_word_not_less
tff(fact_2042_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_2043_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_2044_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_2045_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_2046_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_2047_int__zle__neg,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(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_2048_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ~ ! [N: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nonpos_int_cases
tff(fact_2049_negative__zle__0,axiom,
    ! [Nb: nat] : aa(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_2050_real__add__less__0__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Y),aa(real,real,uminus_uminus(real),Xd)) ) ).

% real_add_less_0_iff
tff(fact_2051_real__0__less__add__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y))
    <=> aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),Xd)),Y) ) ).

% real_0_less_add_iff
tff(fact_2052_real__add__le__0__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Y),aa(real,real,uminus_uminus(real),Xd)) ) ).

% real_add_le_0_iff
tff(fact_2053_real__0__le__add__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y))
    <=> aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),Xd)),Y) ) ).

% real_0_le_add_iff
tff(fact_2054_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A))))
         => ( aa(word(A),$o,ord_less_eq(word(A),Nb),Xd)
           => ( Nb = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_2055_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( Xd != zero_zero(word(A)) )
         => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A)))),Xd) ) ) ).

% word_sub_1_le
tff(fact_2056_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( Xd != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),Xd)
          <=> aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A)))) ) ) ) ).

% word_le_sub1
tff(fact_2057_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A)))),Y)
         => aa(word(A),$o,ord_less_eq(word(A),Xd),Y) ) ) ).

% word_minus_one_le_leq
tff(fact_2058_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),Y)
         => aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ).

% word_le_minus_one_leq
tff(fact_2059_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_2060_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( Xd != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less(word(A),one_one(word(A))),Xd)
          <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A)))) ) ) ) ).

% word_less_sub1
tff(fact_2061_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xd)
        <=> aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A)))),Xd) ) ) ).

% gt0_iff_gem1
tff(fact_2062_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,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_2063_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Xd),Z))
        <=> ( ( Z = zero_zero(word(A)) )
            | aa(word(A),$o,ord_less(word(A),Xd),Z) ) ) ) ).

% sub_wrap
tff(fact_2064_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),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),Xd)),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd)) ) ) ) ).

% word_diff_ls''(1)
tff(fact_2065_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),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),Xd)),Xd)) ) ) ) ).

% word_diff_ls''(2)
tff(fact_2066_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xd: word(A),Z: word(A)] :
          ( aa(word(A),$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),Xd)),Xd)),aa(word(A),word(A),minus_minus(word(A),Z),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd)),Z) ) ) ) ).

% word_l_diffs'(1)
tff(fact_2067_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xd: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd))
           => aa(word(A),$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),Xd)),Xd)),aa(word(A),word(A),minus_minus(word(A),Z),Xd)) ) ) ) ).

% word_l_diffs'(2)
tff(fact_2068_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),W)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd)) ) ) ) ).

% word_diff_ls'(1)
tff(fact_2069_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),W) ) ) ) ).

% word_diff_ls'(2)
tff(fact_2070_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd)),Z) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_2071_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xd: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xd))
           => aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),Xd)) ) ) ) ).

% More_Word.word_l_diffs(2)
tff(fact_2072_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xaa: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),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)),Xd)) ) ) ) ).

% word_diff_ls(2)
tff(fact_2073_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xd)),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)),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$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_2074_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xaa: word(A),Z: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$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)),Xd)),aa(word(A),word(A),minus_minus(word(A),Z),Xd)) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_2075_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xaa: word(A),Xd: word(A),Z: word(A)] :
          ( aa(word(A),$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)),Xd)),aa(word(A),word(A),minus_minus(word(A),Z),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Z)
           => aa(word(A),$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_2076_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xba: word(A),W: word(A),Xaa: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xba)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xba))
           => aa(word(A),$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),Xba)),Xd)),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)),Xd)) ) ) ) ).

% word_plus_mcs(2)
tff(fact_2077_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xba: word(A),Xd: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$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),Xba)),Xd)),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)),Xd))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xba)),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_2078_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Y),Z))
         => ( aa(word(A),$o,ord_less_eq(word(A),Z),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Z)),Y) ) ) ) ).

% word_less_add_right
tff(fact_2079_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
         => ( aa(word(A),$o,ord_less_eq(word(A),Y),Xd)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xd),Y)),Z) ) ) ) ).

% word_less_sub_right
tff(fact_2080_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),C2)) ) ) ) ).

% plus_minus_no_overflow
tff(fact_2081_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),P3)),P3)
        <=> ( P3 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% overflow_plus_one_self
tff(fact_2082_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A)))),Xd)
        <=> ( Xd = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% plus_1_less
tff(fact_2083_max__word__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A))) = zero_zero(word(A)) )
         => ( Xd = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_wrap
tff(fact_2084_less__x__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ( Xd != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A))))
          <=> ( aa(word(A),$o,ord_less(word(A),Y),Xd)
              | ( Y = Xd ) ) ) ) ) ).

% less_x_plus_1
tff(fact_2085_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
         => aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A)))) ) ) ).

% word_add_no_overflow
tff(fact_2086_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),uminus_uminus(word(A)),Y))
         => aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y)) ) ) ).

% no_plus_overflow_neg
tff(fact_2087_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2088_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_2089_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N) )
       => ~ ! [N: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
             => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_2090_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_2091_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)))) ).

% not_zle_0_negative
tff(fact_2092_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_2093_negD,axiom,
    ! [Xd: int] :
      ( aa(int,$o,ord_less(int,Xd),zero_zero(int))
     => ? [N: nat] : Xd = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_2094_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_2095_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Xd: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A)))),B2)
         => ( aa(word(A),$o,ord_less(word(A),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),Xd))
           => aa(word(A),$o,ord_less(word(A),A2),B2) ) ) ) ).

% word_1_0
tff(fact_2096_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N3: word(A),Nb: word(A)] :
          ( aa(word(A),$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_2097_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))))
           => aa(word(A),$o,ord_less(word(A),Xd),Y) ) ) ) ).

% word_leq_minus_one_le
tff(fact_2098_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
         => ( ( Xd != zero_zero(word(A)) )
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A)))),Y) ) ) ) ).

% word_leq_le_minus_one
tff(fact_2099_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xd)
        <=> ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A))))
          <=> aa(word(A),$o,ord_less(word(A),Y),Xd) ) ) ) ).

% le_m1_iff_lt
tff(fact_2100_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,ord_less_eq(word(A),Nb),M)
            & ( Nb != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_2101_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,ord_less_eq(word(A),K),Z)
           => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
             => aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),K)) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_2102_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_2103_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,ord_less_eq(word(A),K),Z)
           => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
             => aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),K)) ) ) ) ) ).

% word_less_nowrapI
tff(fact_2104_word__le__make__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A)] :
          ( ( Y != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
          <=> aa(word(A),$o,ord_less(word(A),Xd),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_2105_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Xd: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A))))
          <=> aa(word(A),$o,ord_less_eq(word(A),Xd),K) ) ) ) ).

% word_Suc_leq
tff(fact_2106_word__Suc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),K: word(A)] :
          ( ( Xd != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A)))),K)
          <=> aa(word(A),$o,ord_less(word(A),Xd),K) ) ) ) ).

% word_Suc_le
tff(fact_2107_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_2108_realpow__square__minus__le,axiom,
    ! [U: real,Xd: real] : aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_2109_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xd: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Xd),Y)))
    <=> ( ( aa(nat,$o,ord_less_eq(nat,Y),Xd)
         => aa(int,$o,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Xd)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,ord_less(nat,Xd),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_2110_real__of__nat__div2,axiom,
    ! [Nb: nat,Xd: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,minus_minus(real,divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),Xd))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,Xd)))) ).

% real_of_nat_div2
tff(fact_2111_real__of__nat__div3,axiom,
    ! [Nb: nat,Xd: nat] : aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),Xd))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,Xd)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_2112_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),Xd)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Y)
           => ( divide_divide(word(A),aa(word(A),word(A),minus_minus(word(A),Xd),Y),Y) = aa(word(A),word(A),minus_minus(word(A),divide_divide(word(A),Xd,Y)),one_one(word(A))) ) ) ) ) ).

% word_div_sub
tff(fact_2113_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_2114_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_2115_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_2116_Bernoulli__inequality,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xd))),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)),Xd)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_2117_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xd: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),Y)),Xd)
         => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),Xd)),Y) ) ) ).

% compl_less_swap2
tff(fact_2118_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xd: A] :
          ( aa(A,$o,ord_less(A,Y),aa(A,A,uminus_uminus(A),Xd))
         => aa(A,$o,ord_less(A,Xd),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_2119_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_12: nat] : aa(nat,$o,P,X_12)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_2120_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A] :
        ? [N: nat] : aa(A,$o,ord_less(A,Xd),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_2121_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_2122_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_2123_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => ? [N: 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),N)),Xd)) ) ) ).

% ex_less_of_nat_mult
tff(fact_2124_space__space_H,axiom,
    ! [T: vEBT_VEBT] : aa(nat,$o,ord_less(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space,T)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T)) ).

% space_space'
tff(fact_2125_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,minus_minus(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) = 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_2126_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,minus_minus(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) = 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_2127_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T))),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_2128_cnt__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(real,$o,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T)),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)),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_2129_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(real,$o,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T)),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_2130_space_H__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T)),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_2131_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ vEBT_invar_vebt(T,zero_zero(nat)) ).

% valid_0_not
tff(fact_2132_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ vEBT_invar_vebt(T,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_2133_deg__not__0,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% deg_not_0
tff(fact_2134_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_2135_space__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space,T)),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_2136_int__if,axiom,
    ! [P: $o,A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),
        $ite((P),A2,B2)) = $ite((P),aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_if
tff(fact_2137_nat__int__comparison_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A2) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_2138_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),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_2139_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xd: nat,Nb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xd)
     => ( vEBT_invar_vebt(Tree,Nb)
       => aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ).

% member_bound
tff(fact_2140_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_2141_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_2142_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Nb: nat] :
      ( ! [N: nat] : aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,H,N)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( aa(nat,A,F2,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_2143_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(Tree,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xd)
       => ( vEBT_V5719532721284313246member(Tree,Xd)
          | vEBT_VEBT_membermima(Tree,Xd) ) ) ) ).

% member_valid_both_member_options
tff(fact_2144_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_2145_member__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(T),Xd)
      <=> member(nat,Xd,vEBT_set_vebt(T)) ) ) ).

% member_correct
tff(fact_2146_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2147_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_2148_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_2149_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_2150_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_2151_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_2152_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,ord_less_eq(nat,Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_2153_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,ord_less_eq(nat,pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_2154_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_2155_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_2156_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_2157_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_2158_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_2159_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_2160_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_2161_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_2162_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xd: A] : neg_numeral_dbl_dec(A,Xd) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Xd)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2163_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(T,Xd)),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,T))),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))) ) ).

% delete_bound_height
tff(fact_2164_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(T,Xd)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% delete_bound_height'
tff(fact_2165_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => ( aa(nat,$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,$o,vEBT_vebt_member(vEBT_vebt_insert(T,Xd)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(T),Y)
              | ( Xd = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_2166_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: 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(T,Deg)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_2167_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K6: real,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),K6)
           => ( aa(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,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(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_2168_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_2169_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),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)) ) ).

% pochhammer_double
tff(fact_2170_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_2171_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_2172_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_2173_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_2174_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_2175_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_2176_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_2177_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_2178_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_2179_log__eq__one,axiom,
    ! [A2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_2180_log__less__cancel__iff,axiom,
    ! [A2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less(real,aa(real,real,log(A2),Xd)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,ord_less(real,Xd),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2181_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(A2),Xd)),one_one(real))
        <=> aa(real,$o,ord_less(real,Xd),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2182_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,log(A2),Xd))
        <=> aa(real,$o,ord_less(real,A2),Xd) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2183_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(A2),Xd)),zero_zero(real))
        <=> aa(real,$o,ord_less(real,Xd),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_2184_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,log(A2),Xd))
        <=> aa(real,$o,ord_less(real,one_one(real)),Xd) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2185_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_2186_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_2187_log__le__cancel__iff,axiom,
    ! [A2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A2),Xd)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2188_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A2),Xd)),one_one(real))
        <=> aa(real,$o,ord_less_eq(real,Xd),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2189_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,log(A2),Xd))
        <=> aa(real,$o,ord_less_eq(real,A2),Xd) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2190_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A2),Xd)),zero_zero(real))
        <=> aa(real,$o,ord_less_eq(real,Xd),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_2191_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,log(A2),Xd))
        <=> aa(real,$o,ord_less_eq(real,one_one(real)),Xd) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_2192_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,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_2193_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => aa(A,$o,ord_less(A,zero_zero(A)),comm_s3205402744901411588hammer(A,Xd,Nb)) ) ) ).

% pochhammer_pos
tff(fact_2194_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,ord_less_eq(nat,Nb),M)
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2195_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,ord_less_eq(nat,Nb),M)
           => ( comm_s3205402744901411588hammer(A,A2,Nb) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2196_log__base__change,axiom,
    ! [A2: real,B2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),Xd) = divide_divide(real,aa(real,real,log(A2),Xd),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_2197_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) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_2198_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)),aa(real,real,log(B2),M)) ) ) ).

% less_log_of_power
tff(fact_2199_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xd: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),comm_s3205402744901411588hammer(A,Xd,Nb)) ) ) ).

% pochhammer_nonneg
tff(fact_2200_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_2201_log__mult,axiom,
    ! [A2: real,Xd: 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)),Xd)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),Xd)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_2202_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,M: real] :
      ( aa(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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),M)) ) ) ).

% le_log_of_power
tff(fact_2203_log__divide,axiom,
    ! [A2: real,Xd: 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)),Xd)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( aa(real,real,log(A2),divide_divide(real,Xd,Y)) = aa(real,real,minus_minus(real,aa(real,real,log(A2),Xd)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_2204_log__base__pow,axiom,
    ! [A2: real,Nb: nat,Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),Nb)),Xd) = divide_divide(real,aa(real,real,log(A2),Xd),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log_base_pow
tff(fact_2205_log__nat__power,axiom,
    ! [Xd: real,B2: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,log(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Xd)) ) ) ).

% log_nat_power
tff(fact_2206_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_2207_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xd: A] : neg_numeral_dbl_inc(A,Xd) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Xd)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_2208_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) = aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ).

% log2_of_power_eq
tff(fact_2209_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_2210_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_2211_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_2212_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_2213_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,ord_less(nat,K3),Nb)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2214_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,aa(real,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_2215_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_2216_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$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_2217_log__of__power__le,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb))
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(real,$o,ord_less_eq(real,aa(real,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_2218_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X7: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [N6: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X7,N6))),K7) )
        <=> ? [N7: nat] :
            ! [N6: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X7,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% lemma_NBseq_def
tff(fact_2219_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X7: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [N6: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X7,N6))),K7) )
        <=> ? [N7: nat] :
            ! [N6: A] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,X7,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% lemma_NBseq_def2
tff(fact_2220_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Nb: nat,Z: A] :
          ( aa(nat,$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_2221_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)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_2222_le__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),M)
     => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_2223_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R2),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2224_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,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_less
tff(fact_2225_log2__of__power__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(real,$o,ord_less_eq(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_le
tff(fact_2226_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_2227_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_2228_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),W))) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_2229_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_2230_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_2231_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_2232_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Xd)),zero_zero(real))
        <=> ( Xd = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2233_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),real_V7770717601297561774m_norm(A,Xd))
        <=> ( Xd != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2234_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_2235_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] :
          ( ( real_V7770717601297561774m_norm(A,Xd) = zero_zero(real) )
        <=> ( Xd = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2236_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2237_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2238_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_2239_pred__member,axiom,
    ! [T: vEBT_VEBT,Xd: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T),Xd,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T),Y)
        & aa(nat,$o,ord_less(nat,Y),Xd)
        & ! [Z5: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T),Z5)
              & aa(nat,$o,ord_less(nat,Z5),Xd) )
           => aa(nat,$o,ord_less_eq(nat,Z5),Y) ) ) ) ).

% pred_member
tff(fact_2240_succ__member,axiom,
    ! [T: vEBT_VEBT,Xd: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T),Xd,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T),Y)
        & aa(nat,$o,ord_less(nat,Xd),Y)
        & ! [Z5: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T),Z5)
              & aa(nat,$o,ord_less(nat,Xd),Z5) )
           => aa(nat,$o,ord_less_eq(nat,Y),Z5) ) ) ) ).

% succ_member
tff(fact_2241_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] : ~ aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_2242_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,Y)) ) ).

% norm_mult
tff(fact_2243_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] : aa(real,$o,ord_less_eq(real,zero_zero(real)),real_V7770717601297561774m_norm(A,Xd)) ) ).

% norm_ge_zero
tff(fact_2244_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,divide_divide(A,A2,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_2245_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: A,Nb: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xd)),Nb) ) ).

% norm_power
tff(fact_2246_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Xd)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) ) ).

% norm_uminus_minus
tff(fact_2247_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_2248_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,divide_divide(A,A2,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2249_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xd: A,R2: real,Y: A,S: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),R2)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S)) ) ) ) ).

% norm_mult_less
tff(fact_2250_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xd: A,Y: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_2251_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A,R2: real,Y: A,S: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),R2)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S)) ) ) ) ).

% norm_add_less
tff(fact_2252_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A,Y: A,E: real] :
          ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y))),E) ) ) ).

% norm_triangle_lt
tff(fact_2253_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,S: real] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,A2)),R2)
         => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,B2)),S)
           => aa(real,$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),R2),S)) ) ) ) ).

% norm_triangle_mono
tff(fact_2254_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A,Y: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_2255_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A,Y: A,E: real] :
          ( aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y))),E) ) ) ).

% norm_triangle_le
tff(fact_2256_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( aa(real,$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,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_2257_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xd),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,Xd),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2258_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xd: A,Nb: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xd)),Nb)) ) ).

% norm_power_ineq
tff(fact_2259_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$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_2260_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_2261_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D)))),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),D)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2262_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,Xd) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_2263_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,W)),one_one(real))
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Z),W)))) ) ) ) ).

% norm_power_diff
tff(fact_2264_inrange,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(set(nat),$o,ord_less_eq(set(nat),vEBT_VEBT_set_vebt(T)),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_2265_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_2266_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_2267_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T,Xd)),Xd) ) ) ).

% valid_insert_both_member_options_add
tff(fact_2268_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => ( aa(nat,$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,$o,vEBT_V8194947554948674370ptions(T),Xd)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T,Y)),Xd) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_2269_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( aa(A,A,artanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_2270_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xd)
       => aa(nat,$o,vEBT_vebt_member(T),Xd) ) ) ).

% valid_member_both_member_options
tff(fact_2271_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xd)
      <=> aa(nat,$o,vEBT_vebt_member(T),Xd) ) ) ).

% both_member_options_equiv_member
tff(fact_2272_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,Xd: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xd)
    <=> ( vEBT_V5719532721284313246member(T,Xd)
        | vEBT_VEBT_membermima(T,Xd) ) ) ).

% both_member_options_def
tff(fact_2273_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_2274_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_2275_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_2276_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xd)),zero_zero(int))
        <=> aa(A,$o,ord_less_eq(A,Xd),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_2277_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xd)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less_eq(A,Xd),aa(num,A,numeral_numeral(A),V)) ) ) ).

% ceiling_le_numeral
tff(fact_2278_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),Xd) ) ) ).

% zero_less_ceiling
tff(fact_2279_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),V)),Xd) ) ) ).

% numeral_less_ceiling
tff(fact_2280_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xd)),one_one(int))
        <=> aa(A,$o,ord_less_eq(A,Xd),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_2281_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,one_one(int)),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),Xd) ) ) ).

% one_le_ceiling
tff(fact_2282_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xd)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_2283_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xd)),one_one(int))
        <=> aa(A,$o,ord_less_eq(A,Xd),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_2284_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,one_one(int)),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,one_one(A)),Xd) ) ) ).

% one_less_ceiling
tff(fact_2285_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_2286_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xd),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xd)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_2287_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xd)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_2288_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: num,Nb: nat] : archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xd)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb) ) ).

% ceiling_numeral_power
tff(fact_2289_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xd),one_one(A))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xd)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_2290_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xd)),zero_zero(int))
        <=> aa(A,$o,ord_less_eq(A,Xd),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_2291_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),Xd) ) ) ).

% zero_le_ceiling
tff(fact_2292_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,uminus_uminus(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_2293_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xd)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less_eq(A,Xd),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_2294_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))),Xd) ) ) ).

% numeral_le_ceiling
tff(fact_2295_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xd)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less_eq(A,Xd),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2296_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xd) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2297_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(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_2298_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xd)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less_eq(A,Xd),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_2299_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xd))
        <=> 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))),Xd) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2300_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less(set(A),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( ( ~ aa(A,$o,ord_less_eq(A,A2),B2)
              | ( aa(A,$o,ord_less_eq(A,C2),A2)
                & aa(A,$o,ord_less_eq(A,B2),D)
                & ( aa(A,$o,ord_less(A,C2),A2)
                  | aa(A,$o,ord_less(A,B2),D) ) ) )
            & aa(A,$o,ord_less_eq(A,C2),D) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2301_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xd)),archimedean_ceiling(A,Y))
         => aa(A,$o,ord_less(A,Xd),Y) ) ) ).

% ceiling_less_cancel
tff(fact_2302_ex__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M4: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M4),Nb)
          & aa(nat,$o,P,M4) )
    <=> ? [X5: nat] :
          ( member(nat,X5,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X5) ) ) ).

% ex_nat_less
tff(fact_2303_all__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M4),Nb)
         => aa(nat,$o,P,M4) )
    <=> ! [X5: nat] :
          ( member(nat,X5,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X5) ) ) ).

% all_nat_less
tff(fact_2304_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: A] : aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xd)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_2305_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(int,$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_2306_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,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,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archimedean_ceiling(real,aa(real,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_2307_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archimedean_ceiling(real,aa(real,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,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),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_2308_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(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,aa(real,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,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_2309_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xd: A] : aa(A,A,artanh(A),Xd) = divide_divide(A,aa(A,A,ln_ln(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Xd),aa(A,A,minus_minus(A,one_one(A)),Xd))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% artanh_def
tff(fact_2310_log__ceil__idem,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),Xd)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,Xd)))) ) ) ).

% log_ceil_idem
tff(fact_2311_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_2312_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),Xd: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,Xd,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,ord_less(nat,Xd),Y)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => ( aa(nat,$o,ord_less(nat,Xd),X5)
             => aa(nat,$o,ord_less_eq(nat,Y),X5) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_2313_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),Xd: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,Xd,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,ord_less(nat,Y),Xd)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => ( aa(nat,$o,ord_less(nat,X5),Xd)
             => aa(nat,$o,ord_less_eq(nat,X5),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_2314_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_2315_ln__less__cancel__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xd)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,ord_less(real,Xd),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2316_ln__inj__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),Xd) = aa(real,real,ln_ln(real),Y) )
        <=> ( Xd = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2317_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xd)) = Xd )
        <=> ? [N6: int] : Xd = aa(int,A,ring_1_of_int(A),N6) ) ) ).

% of_int_ceiling_cancel
tff(fact_2318_of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = zero_zero(A) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_eq_0_iff
tff(fact_2319_of__int__0__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( zero_zero(A) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_0_eq_iff
tff(fact_2320_of__int__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),zero_zero(int)) = zero_zero(A) ) ) ).

% of_int_0
tff(fact_2321_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_2322_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_2323_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_2324_ln__le__cancel__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xd)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2325_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_2326_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_2327_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_2328_ln__less__zero__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xd)),zero_zero(real))
      <=> aa(real,$o,ord_less(real,Xd),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_2329_ln__gt__zero__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),Xd))
      <=> aa(real,$o,ord_less(real,one_one(real)),Xd) ) ) ).

% ln_gt_zero_iff
tff(fact_2330_ln__eq__zero__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( ( aa(real,real,ln_ln(real),Xd) = zero_zero(real) )
      <=> ( Xd = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_2331_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_2332_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2333_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_2334_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,Xd: 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),Xd) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = Xd ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_2335_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xd: int,B2: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Xd) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) )
        <=> ( Xd = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_2336_ln__le__zero__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xd)),zero_zero(real))
      <=> aa(real,$o,ord_less_eq(real,Xd),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_2337_ln__ge__zero__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),Xd))
      <=> aa(real,$o,ord_less_eq(real,one_one(real)),Xd) ) ) ).

% ln_ge_zero_iff
tff(fact_2338_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xd)),Z) ) ).

% ceiling_add_of_int
tff(fact_2339_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,ord_less_eq(int,Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_2340_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_2341_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,ord_less_eq(int,Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_le_numeral_iff
tff(fact_2342_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_2343_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_2344_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_2345_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_2346_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_2347_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,ord_less_eq(int,Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_2348_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_2349_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_2350_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_2351_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xd: num,Nb: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xd)),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),Xd)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2352_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xd: 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),Xd)),Nb) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2353_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: int,B2: int,W: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Xd)),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_eq(int,Xd),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2354_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xd: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xd))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xd) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2355_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: int,B2: int,W: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Xd)),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,Xd),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2356_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xd: int] :
          ( aa(A,$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),Xd))
        <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xd) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2357_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: num,Nb: nat,A2: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xd)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2358_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xd: num,Nb: nat] :
          ( aa(A,$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),Xd)),Nb))
        <=> aa(int,$o,ord_less_eq(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2359_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: num,Nb: nat,A2: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xd)),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),Xd)),Nb)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2360_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xd: num,Nb: nat] :
          ( aa(A,$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),Xd)),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),Xd)),Nb)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2361_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xd: 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),Xd))),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),Xd))),Nb) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2362_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xd: 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),Xd))),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),Xd))),Nb) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2363_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xd: num,Nb: nat] :
          ( aa(A,$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),Xd))),Nb))
        <=> aa(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),Xd))),Nb)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2364_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: num,Nb: nat,A2: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xd))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xd))),Nb)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2365_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: num,Nb: nat,A2: int] :
          ( aa(A,$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),Xd))),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),Xd))),Nb)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2366_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xd: num,Nb: nat] :
          ( aa(A,$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),Xd))),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),Xd))),Nb)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2367_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A] :
        ? [Z2: int] : aa(A,$o,ord_less(A,Xd),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_less_of_int
tff(fact_2368_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A] :
        ? [Z2: int] : aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z2)),Xd) ) ).

% ex_of_int_less
tff(fact_2369_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xd)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xd)) ) ).

% mult_of_int_commute
tff(fact_2370_ln__less__self,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xd)),Xd) ) ).

% ln_less_self
tff(fact_2371_log__def,axiom,
    ! [A2: real,Xd: real] : aa(real,real,log(A2),Xd) = divide_divide(real,aa(real,real,ln_ln(real),Xd),aa(real,real,ln_ln(real),A2)) ).

% log_def
tff(fact_2372_ln__bound,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xd)),Xd) ) ).

% ln_bound
tff(fact_2373_ln__gt__zero__imp__gt__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),Xd))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => aa(real,$o,ord_less(real,one_one(real)),Xd) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_2374_ln__less__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xd)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_2375_ln__gt__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),Xd)) ) ).

% ln_gt_zero
tff(fact_2376_ln__ge__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),Xd)) ) ).

% ln_ge_zero
tff(fact_2377_bset_I1_J,axiom,
    ! [D3: int,B4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,B4)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,B4)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3))
                & aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_2378_bset_I2_J,axiom,
    ! [D3: int,B4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,B4)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,B4)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3))
                | aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_2379_aset_I1_J,axiom,
    ! [D3: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,A3)
                 => ( X != aa(int,int,minus_minus(int,Xb),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,A3)
                   => ( X != aa(int,int,minus_minus(int,Xb),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_2380_aset_I2_J,axiom,
    ! [D3: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,A3)
                 => ( X != aa(int,int,minus_minus(int,Xb),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,A3)
                   => ( X != aa(int,int,minus_minus(int,Xb),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_2381_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,A2: int] :
          ( aa(A,$o,ord_less_eq(A,Xd),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xd)),A2) ) ) ).

% ceiling_le
tff(fact_2382_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xd: A] :
          ( aa(int,$o,ord_less(int,Z),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),Xd) ) ) ).

% less_ceiling_iff
tff(fact_2383_real__of__int__div4,axiom,
    ! [Nb: int,Xd: int] : aa(real,$o,ord_less_eq(real,aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xd))),divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xd))) ).

% real_of_int_div4
tff(fact_2384_real__of__int__div,axiom,
    ! [D: int,Nb: int] :
      ( aa(int,$o,dvd_dvd(int,D),Nb)
     => ( aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,D)) = divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),D)) ) ) ).

% real_of_int_div
tff(fact_2385_ln__ge__zero__imp__ge__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),Xd))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => aa(real,$o,ord_less_eq(real,one_one(real)),Xd) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2386_ln__add__one__self__le__self,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd))),Xd) ) ).

% ln_add_one_self_le_self
tff(fact_2387_ln__mult,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$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),Xd),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xd)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2388_ln__eq__minus__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( ( aa(real,real,ln_ln(real),Xd) = aa(real,real,minus_minus(real,Xd),one_one(real)) )
       => ( Xd = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_2389_ln__div,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),divide_divide(real,Xd,Y)) = aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),Xd)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_2390_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_2391_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_2392_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A] :
        ? [X: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),X)),Xd)
          & aa(A,$o,ord_less(A,Xd),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int))))
          & ! [Y3: int] :
              ( ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y3)),Xd)
                & aa(A,$o,ord_less(A,Xd),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int)))) )
             => ( Y3 = X ) ) ) ) ).

% floor_exists1
tff(fact_2393_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A] :
        ? [Z2: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z2)),Xd)
          & aa(A,$o,ord_less(A,Xd),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))) ) ) ).

% floor_exists
tff(fact_2394_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_2395_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_2396_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2) ) ).

% of_int_ceiling_diff_one_le
tff(fact_2397_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,Xd: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(int,A,ring_1_of_int(A),Xd))
        <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),Nb)),Xd) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2398_aset_I10_J,axiom,
    ! [D: int,D3: int,A3: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T)) ) ) ) ).

% aset(10)
tff(fact_2399_aset_I9_J,axiom,
    ! [D: int,D3: int,A3: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T)) ) ) ) ).

% aset(9)
tff(fact_2400_bset_I10_J,axiom,
    ! [D: int,D3: int,B4: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X4),D3)),T)) ) ) ) ).

% bset(10)
tff(fact_2401_bset_I9_J,axiom,
    ! [D: int,D3: int,B4: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X4),D3)),T)) ) ) ) ).

% bset(9)
tff(fact_2402_int__le__real__less,axiom,
    ! [Nb: int,M: int] :
      ( aa(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_2403_int__less__real__le,axiom,
    ! [Nb: int,M: int] :
      ( aa(int,$o,ord_less(int,Nb),M)
    <=> aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_2404_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(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),divide_divide(int,aa(int,int,uminus_uminus(int),A2),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_2405_real__of__int__div__aux,axiom,
    ! [Xd: int,D: int] : divide_divide(real,aa(int,real,ring_1_of_int(real),Xd),aa(int,real,ring_1_of_int(real),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),divide_divide(int,Xd,D))),divide_divide(real,aa(int,real,ring_1_of_int(real),modulo_modulo(int,Xd,D)),aa(int,real,ring_1_of_int(real),D))) ).

% real_of_int_div_aux
tff(fact_2406_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_2407_ln__le__minus__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xd)),aa(real,real,minus_minus(real,Xd),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_2408_ln__diff__le,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),Xd)),aa(real,real,ln_ln(real),Y))),divide_divide(real,aa(real,real,minus_minus(real,Xd),Y),Y)) ) ) ).

% ln_diff_le
tff(fact_2409_ln__add__one__self__le__self2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd))),Xd) ) ).

% ln_add_one_self_le_self2
tff(fact_2410_ln__realpow,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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),Xd)) ) ) ).

% ln_realpow
tff(fact_2411_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xd))),one_one(A))),Xd)
          & aa(A,$o,ord_less_eq(A,Xd),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xd))) ) ) ).

% ceiling_correct
tff(fact_2412_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xd: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xd)
         => ( aa(A,$o,ord_less_eq(A,Xd),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xd) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_2413_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,A2: int] :
          ( ( archimedean_ceiling(A,Xd) = A2 )
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xd)
            & aa(A,$o,ord_less_eq(A,Xd),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_2414_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,T))
        <=> ! [I2: int] :
              ( ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I2)),one_one(A))),T)
                & aa(A,$o,ord_less_eq(A,T),aa(int,A,ring_1_of_int(A),I2)) )
             => aa(int,$o,P,I2) ) ) ) ).

% ceiling_split
tff(fact_2415_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Z: int] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xd)),Z)
        <=> aa(A,$o,ord_less_eq(A,Xd),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_2416_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xd: A] :
          ( aa(int,$o,ord_less_eq(int,Z),archimedean_ceiling(A,Xd))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xd) ) ) ).

% le_ceiling_iff
tff(fact_2417_bset_I3_J,axiom,
    ! [D3: int,T: int,B4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,T),one_one(int)),B4)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X4 = T )
             => ( aa(int,int,minus_minus(int,X4),D3) = T ) ) ) ) ) ).

% bset(3)
tff(fact_2418_bset_I4_J,axiom,
    ! [D3: int,T: int,B4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,T,B4)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X4 != T )
             => ( aa(int,int,minus_minus(int,X4),D3) != T ) ) ) ) ) ).

% bset(4)
tff(fact_2419_bset_I5_J,axiom,
    ! [D3: int,B4: set(int),T: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less(int,X4),T)
           => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,X4),D3)),T) ) ) ) ).

% bset(5)
tff(fact_2420_bset_I7_J,axiom,
    ! [D3: int,T: int,B4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,T,B4)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less(int,T),X4)
             => aa(int,$o,ord_less(int,T),aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ).

% bset(7)
tff(fact_2421_aset_I3_J,axiom,
    ! [D3: int,T: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int)),A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( X4 = T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3) = T ) ) ) ) ) ).

% aset(3)
tff(fact_2422_aset_I4_J,axiom,
    ! [D3: int,T: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,T,A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( X4 != T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3) != T ) ) ) ) ) ).

% aset(4)
tff(fact_2423_aset_I5_J,axiom,
    ! [D3: int,T: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,T,A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less(int,X4),T)
             => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T) ) ) ) ) ).

% aset(5)
tff(fact_2424_aset_I7_J,axiom,
    ! [D3: int,A3: set(int),T: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less(int,T),X4)
           => aa(int,$o,ord_less(int,T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ).

% aset(7)
tff(fact_2425_periodic__finite__ex,axiom,
    ! [D: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P,X)
          <=> aa(int,$o,P,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
       => ( ? [X_13: int] : aa(int,$o,P,X_13)
        <=> ? [X5: int] :
              ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D))
              & aa(int,$o,P,X5) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2426_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),R2: word(A),Y: word(A),S: word(A)] :
          ( aa(set(word(A)),$o,ord_less_eq(set(word(A)),set_or1337092689740270186AtMost(word(A),Xd,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)),Xd),R2)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S)),one_one(word(A)))))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),R2)),one_one(word(A))))
           => ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S)),one_one(word(A))))
             => ( ( S != zero_zero(word(A)) )
               => aa(word(A),$o,ord_less_eq(word(A),R2),S) ) ) ) ) ) ).

% word_subset_less
tff(fact_2427_real__of__int__div2,axiom,
    ! [Nb: int,Xd: int] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,minus_minus(real,divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xd))),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xd)))) ).

% real_of_int_div2
tff(fact_2428_real__of__int__div3,axiom,
    ! [Nb: int,Xd: int] : aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xd))),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xd)))),one_one(real)) ).

% real_of_int_div3
tff(fact_2429_ln__one__minus__pos__upper__bound,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),Xd))),aa(real,real,uminus_uminus(real),Xd)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2430_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xd: 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)),Xd)
             => ( aa(real,real,log(A2),Xd) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),B2),aa(real,real,ln_ln(real),A2))),aa(real,real,log(B2),Xd)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2431_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,ord_less_eq(A,P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P3,Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_2432_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_2433_bset_I6_J,axiom,
    ! [D3: int,B4: set(int),T: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less_eq(int,X4),T)
           => aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,X4),D3)),T) ) ) ) ).

% bset(6)
tff(fact_2434_bset_I8_J,axiom,
    ! [D3: int,T: int,B4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,T),one_one(int)),B4)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less_eq(int,T),X4)
             => aa(int,$o,ord_less_eq(int,T),aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ).

% bset(8)
tff(fact_2435_aset_I6_J,axiom,
    ! [D3: int,T: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int)),A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less_eq(int,X4),T)
             => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T) ) ) ) ) ).

% aset(6)
tff(fact_2436_aset_I8_J,axiom,
    ! [D3: int,A3: set(int),T: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less_eq(int,T),X4)
           => aa(int,$o,ord_less_eq(int,T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ).

% aset(8)
tff(fact_2437_cpmi,axiom,
    ! [D3: int,P: fun(int,$o),P2: fun(int,$o),B4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa(int,$o,ord_less(int,X),Z3)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb: int] :
                      ( member(int,Xb,B4)
                     => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X),D3)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P2,X)
              <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P2,X5) )
                | ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,B4)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2438_cppi,axiom,
    ! [D3: int,P: fun(int,$o),P2: fun(int,$o),A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa(int,$o,ord_less(int,Z3),X)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb: int] :
                      ( member(int,Xb,A3)
                     => ( X != aa(int,int,minus_minus(int,Xb),Xa) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D3)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P2,X)
              <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P2,X5) )
                | ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,A3)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2439_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$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,divide_divide(A,P3,Q2)))),one_one(A))),Q2)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_2440_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int,Xd: A] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Nb)),Xd)
         => ( aa(A,$o,ord_less_eq(A,Xd),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,Xd) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_2441_ln__one__plus__pos__lower__bound,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,Xd),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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)),Xd))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2442_member__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_m_e_m_b_e_r(T,Xd)),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,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_2443_ln__one__minus__pos__lower__bound,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,uminus_uminus(real),Xd)),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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),Xd))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2444_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
     => ( ( A3 != B4 )
       => aa(set(A),$o,ord_less(set(A),A3),B4) ) ) ).

% psubsetI
tff(fact_2445_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_2446_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_2447_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_2448_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_s_u_c_c(T,Xd)),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,T))),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_2449_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_p_r_e_d(T,Xd)),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,T))),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_2450_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_2451_word__of__int__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),zero_zero(int)) = zero_zero(word(A)) ) ) ).

% word_of_int_0
tff(fact_2452_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_2453_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_2454_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_2455_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_2456_psubsetD,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C2: A] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
     => ( member(A,C2,A3)
       => member(A,C2,B4) ) ) ).

% psubsetD
tff(fact_2457_psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
     => ( aa(set(A),$o,ord_less(set(A),B4),C6)
       => aa(set(A),$o,ord_less(set(A),A3),C6) ) ) ).

% psubset_trans
tff(fact_2458_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
     => ? [B3: A] : member(A,B3,aa(set(A),set(A),minus_minus(set(A),B4),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_2459_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_2460_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_2461_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
     => ~ ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
         => aa(set(A),$o,ord_less_eq(set(A),B4),A3) ) ) ).

% psubsetE
tff(fact_2462_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
        & ( A3 != B4 ) ) ) ).

% psubset_eq
tff(fact_2463_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
     => aa(set(A),$o,ord_less_eq(set(A),A3),B4) ) ).

% psubset_imp_subset
tff(fact_2464_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
     => ( aa(set(A),$o,ord_less_eq(set(A),B4),C6)
       => aa(set(A),$o,ord_less(set(A),A3),C6) ) ) ).

% psubset_subset_trans
tff(fact_2465_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
        & ~ aa(set(A),$o,ord_less_eq(set(A),B4),A3) ) ) ).

% subset_not_subset_eq
tff(fact_2466_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C6: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
     => ( aa(set(A),$o,ord_less(set(A),B4),C6)
       => aa(set(A),$o,ord_less(set(A),A3),C6) ) ) ).

% subset_psubset_trans
tff(fact_2467_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
    <=> ( aa(set(A),$o,ord_less(set(A),A3),B4)
        | ( A3 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2468_insersimp,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T),X_1)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(T,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_2469_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(T,Xd)),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,T))),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_2470_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_2471_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
       => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd))),Xd))),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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2472_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: int] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,Xd),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,Xd) = Y ) ) ) ) ).

% round_unique
tff(fact_2473_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( vEBT_invar_vebt(T,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,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(T,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_2474_tanh__ln__real,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),Xd)) = divide_divide(real,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2475_obtain__set__pred,axiom,
    ! [Z: nat,Xd: nat,A3: set(nat)] :
      ( aa(nat,$o,ord_less(nat,Z),Xd)
     => ( vEBT_VEBT_min_in_set(A3,Z)
       => ( aa(set(nat),$o,finite_finite2(nat),A3)
         => ? [X_1: nat] : vEBT_is_pred_in_set(A3,Xd,X_1) ) ) ) ).

% obtain_set_pred
tff(fact_2476_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(set(nat),$o,finite_finite2(nat),vEBT_VEBT_set_vebt(T)) ) ).

% set_vebt_finite
tff(fact_2477_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_succ_in_set(Xs,A2,X_1)
     => ( aa(set(nat),$o,finite_finite2(nat),Xs)
       => ~ ? [X4: nat] :
              ( member(nat,X4,Xs)
              & aa(nat,$o,ord_less(nat,A2),X4) ) ) ) ).

% succ_none_empty
tff(fact_2478_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_pred_in_set(Xs,A2,X_1)
     => ( aa(set(nat),$o,finite_finite2(nat),Xs)
       => ~ ? [X4: nat] :
              ( member(nat,X4,Xs)
              & aa(nat,$o,ord_less(nat,X4),A2) ) ) ) ).

% pred_none_empty
tff(fact_2479_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : abs_abs(A,abs_abs(A,A2)) = abs_abs(A,A2) ) ).

% abs_abs
tff(fact_2480_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : abs_abs(A,abs_abs(A,A2)) = abs_abs(A,A2) ) ).

% abs_idempotent
tff(fact_2481_obtain__set__succ,axiom,
    ! [Xd: nat,Z: nat,A3: set(nat),B4: set(nat)] :
      ( aa(nat,$o,ord_less(nat,Xd),Z)
     => ( vEBT_VEBT_max_in_set(A3,Z)
       => ( aa(set(nat),$o,finite_finite2(nat),B4)
         => ( ( A3 = B4 )
           => ? [X_1: nat] : vEBT_is_succ_in_set(A3,Xd,X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_2482_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2483_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( abs_abs(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2484_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = abs_abs(A,A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2485_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_2486_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : abs_abs(A,aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ).

% abs_numeral
tff(fact_2487_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A2)),abs_abs(A,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A2)),abs_abs(A,B2)) ) ).

% abs_add_abs
tff(fact_2488_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_2489_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A2)),abs_abs(A,A2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% abs_mult_self_eq
tff(fact_2490_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : abs_abs(A,divide_divide(A,A2,B2)) = divide_divide(A,abs_abs(A,A2),abs_abs(A,B2)) ) ).

% abs_divide
tff(fact_2491_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : abs_abs(A,aa(A,A,uminus_uminus(A),A2)) = abs_abs(A,A2) ) ).

% abs_minus
tff(fact_2492_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : abs_abs(A,aa(A,A,uminus_uminus(A),A2)) = abs_abs(A,A2) ) ).

% abs_minus_cancel
tff(fact_2493_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,M),abs_abs(A,K))
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% dvd_abs_iff
tff(fact_2494_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,abs_abs(A,M)),K)
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% abs_dvd_iff
tff(fact_2495_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : 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_2496_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : abs_abs(A,aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% abs_bool_eq
tff(fact_2497_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_2498_tanh__real__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa(real,real,tanh(real),Xd) = zero_zero(real) )
    <=> ( Xd = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_2499_tanh__real__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,tanh(real),Xd)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,ord_less(real,Xd),Y) ) ).

% tanh_real_less_iff
tff(fact_2500_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( abs_abs(A,A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_2501_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A2)),A2)
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A2) ) ) ).

% abs_le_self_iff
tff(fact_2502_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A2)),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2503_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),abs_abs(A,A2))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2504_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : 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_2505_abs__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( abs_abs(A,aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).

% abs_neg_one
tff(fact_2506_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_2507_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : abs_abs(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb)) = abs_abs(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% abs_power_minus
tff(fact_2508_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2509_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_2510_tanh__real__pos__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,tanh(real),Xd))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xd) ) ).

% tanh_real_pos_iff
tff(fact_2511_tanh__real__neg__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,tanh(real),Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xd),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_2512_tanh__real__nonpos__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,tanh(real),Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xd),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_2513_tanh__real__nonneg__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,tanh(real),Xd))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd) ) ).

% tanh_real_nonneg_iff
tff(fact_2514_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2515_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,divide_divide(A,A2,abs_abs(A,B2))),zero_zero(A))
        <=> ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2516_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),divide_divide(A,A2,abs_abs(A,B2)))
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2517_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),zero_zero(A))
         => ( abs_abs(A,A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_2518_artanh__minus__real,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xd)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xd)) ) ) ).

% artanh_minus_real
tff(fact_2519_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),abs_abs(A,A2)),Nb))
        <=> ( ( A2 != zero_zero(A) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2520_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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_2521_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(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_2522_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_2523_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),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_2524_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( abs_abs(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2525_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A2)),B2)
         => aa(A,$o,ord_less_eq(A,A2),B2) ) ) ).

% abs_le_D1
tff(fact_2526_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,ord_less_eq(A,A2),abs_abs(A,A2)) ) ).

% abs_ge_self
tff(fact_2527_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2528_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: 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),abs_abs(A,A2)),abs_abs(A,B2)) ) ).

% abs_mult
tff(fact_2529_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : abs_abs(A,aa(A,A,minus_minus(A,A2),B2)) = abs_abs(A,aa(A,A,minus_minus(A,B2),A2)) ) ).

% abs_minus_commute
tff(fact_2530_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xd: A,Y: A] :
          ( ( abs_abs(A,Xd) = abs_abs(A,Y) )
        <=> ( ( Xd = Y )
            | ( Xd = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2531_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : 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),abs_abs(A,A2)),Nb) ) ).

% power_abs
tff(fact_2532_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( abs_abs(A,L) = abs_abs(A,K) )
         => aa(A,$o,dvd_dvd(A,L),K) ) ) ).

% dvd_if_abs_eq
tff(fact_2533_finite__nat__set__iff__bounded,axiom,
    ! [N2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N2)
    <=> ? [M4: nat] :
        ! [X5: nat] :
          ( member(nat,X5,N2)
         => aa(nat,$o,ord_less(nat,X5),M4) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_2534_bounded__nat__set__is__finite,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( ! [X: nat] :
          ( member(nat,X,N2)
         => aa(nat,$o,ord_less(nat,X),Nb) )
     => aa(set(nat),$o,finite_finite2(nat),N2) ) ).

% bounded_nat_set_is_finite
tff(fact_2535_finite__set__decode,axiom,
    ! [Nb: nat] : aa(set(nat),$o,finite_finite2(nat),nat_set_decode(Nb)) ).

% finite_set_decode
tff(fact_2536_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),abs_abs(A,A2)) ) ).

% abs_ge_zero
tff(fact_2537_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,abs_abs(A,A2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2538_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( abs_abs(A,A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2539_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,ord_less_eq(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),abs_abs(A,A2)),abs_abs(A,B2))) ) ).

% abs_triangle_ineq
tff(fact_2540_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,ord_less(A,abs_abs(A,A2)),C2)
         => ( aa(A,$o,ord_less(A,abs_abs(A,B2)),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A2)),abs_abs(A,B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D)) ) ) ) ).

% abs_mult_less
tff(fact_2541_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,abs_abs(A,A2)),abs_abs(A,B2))),abs_abs(A,aa(A,A,minus_minus(A,B2),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2542_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,abs_abs(A,A2)),abs_abs(A,B2)))),abs_abs(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_2543_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,abs_abs(A,A2)),abs_abs(A,B2))),abs_abs(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_2544_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( abs_abs(A,divide_divide(A,A2,B2)) = divide_divide(A,abs_abs(A,A2),abs_abs(A,B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2545_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),abs_abs(A,A2)) ) ).

% abs_ge_minus_self
tff(fact_2546_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A2)),B2)
        <=> ( aa(A,$o,ord_less_eq(A,A2),B2)
            & aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_2547_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A2)),B2)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2) ) ) ).

% abs_le_D2
tff(fact_2548_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2)
           => aa(A,$o,ord_less_eq(A,abs_abs(A,A2)),B2) ) ) ) ).

% abs_leI
tff(fact_2549_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(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_2550_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_2551_tanh__real__lt__1,axiom,
    ! [Xd: real] : aa(real,$o,ord_less(real,aa(real,real,tanh(real),Xd)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2552_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xd: A] :
          ( ! [E2: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),E2)
             => aa(A,$o,ord_less_eq(A,abs_abs(A,Xd)),E2) )
         => ( Xd = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2553_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,Y)),Xd) = abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xd)) ) ) ) ).

% abs_mult_pos
tff(fact_2554_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
              | aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) )
            & ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
              | aa(A,$o,ord_less_eq(A,B2),zero_zero(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),abs_abs(A,A2)),abs_abs(A,B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2555_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),abs_abs(A,A2))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2556_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( abs_abs(A,A2) = B2 )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2557_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = abs_abs(A,B2) )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2558_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A2)),Nb)) ) ).

% zero_le_power_abs
tff(fact_2559_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xd: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( divide_divide(A,abs_abs(A,Xd),Y) = abs_abs(A,divide_divide(A,Xd,Y)) ) ) ) ).

% abs_div_pos
tff(fact_2560_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X4: A] :
          abs_abs(A,X4) = $ite(aa(A,$o,ord_less(A,X4),zero_zero(A)),aa(A,A,uminus_uminus(A),X4),X4) ) ).

% abs_if_raw
tff(fact_2561_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( abs_abs(A,A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_neg
tff(fact_2562_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: 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_2563_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,aa(A,A,minus_minus(A,A2),C2))),abs_abs(A,aa(A,A,minus_minus(A,B2),D)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2564_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A2)),abs_abs(A,B2))) ) ).

% abs_triangle_ineq4
tff(fact_2565_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,A2: A,R2: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,Xd),A2))),R2)
        <=> ( aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A2),R2)),Xd)
            & aa(A,$o,ord_less_eq(A,Xd),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2566_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,A2: A,R2: A] :
          ( aa(A,$o,ord_less(A,abs_abs(A,aa(A,A,minus_minus(A,Xd),A2))),R2)
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A2),R2)),Xd)
            & aa(A,$o,ord_less(A,Xd),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2567_abs__real__def,axiom,
    ! [A2: real] :
      abs_abs(real,A2) = $ite(aa(real,$o,ord_less(real,A2),zero_zero(real)),aa(real,real,uminus_uminus(real),A2),A2) ).

% abs_real_def
tff(fact_2568_lemma__interval__lt,axiom,
    ! [A2: real,Xd: real,B2: real] :
      ( aa(real,$o,ord_less(real,A2),Xd)
     => ( aa(real,$o,ord_less(real,Xd),B2)
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [Y3: real] :
                ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xd),Y3))),D2)
               => ( aa(real,$o,ord_less(real,A2),Y3)
                  & aa(real,$o,ord_less(real,Y3),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2569_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite2(nat),N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2570_tanh__real__gt__neg1,axiom,
    ! [Xd: real] : aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),Xd)) ).

% tanh_real_gt_neg1
tff(fact_2571_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] : aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),abs_abs(A,Xd))) ) ).

% abs_add_one_gt_zero
tff(fact_2572_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xd: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,aa(int,A,ring_1_of_int(A),Nb))),Xd)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,ord_less_eq(A,one_one(A)),Xd) ) ) ) ).

% of_int_leD
tff(fact_2573_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xd: A] :
          ( aa(A,$o,ord_less(A,abs_abs(A,aa(int,A,ring_1_of_int(A),Nb))),Xd)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,ord_less(A,one_one(A)),Xd) ) ) ) ).

% of_int_lessD
tff(fact_2574_lemma__interval,axiom,
    ! [A2: real,Xd: real,B2: real] :
      ( aa(real,$o,ord_less(real,A2),Xd)
     => ( aa(real,$o,ord_less(real,Xd),B2)
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [Y3: real] :
                ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xd),Y3))),D2)
               => ( aa(real,$o,ord_less_eq(real,A2),Y3)
                  & aa(real,$o,ord_less_eq(real,Y3),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_2575_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,Xd))),Xd))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_2576_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Nb: int] :
          ( aa(A,$o,ord_less(A,abs_abs(A,aa(A,A,minus_minus(A,Xd),aa(int,A,ring_1_of_int(A),Nb)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,Xd) = Nb ) ) ) ).

% round_unique'
tff(fact_2577_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,Xd)),abs_abs(A,Y))
        <=> aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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_2578_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( abs_abs(A,Xd) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2579_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),abs_abs(A,A2)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% power_even_abs
tff(fact_2580_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xd: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,ord_less_eq(A,abs_abs(A,Xd)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2581_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,ord_less_eq(A,abs_abs(A,Xd)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2582_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,ord_less(A,abs_abs(A,Xd)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2583_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,$o,ord_less_eq(A,abs_abs(A,A2)),abs_abs(A,B2))
           => aa(A,$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_2584_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_p_r_e_d2(T,Xd)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% pred_bound_height'
tff(fact_2585_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_s_u_c_c2(T,Xd)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% succ'_bound_height
tff(fact_2586_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd))),Xd))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2587_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,Xd))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_2588_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,Xd),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,Xd))) ) ).

% of_int_round_ge
tff(fact_2589_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,Xd),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,Xd))) ) ).

% of_int_round_gt
tff(fact_2590_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd))),Xd))),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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2591_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xd: A] :
          ( ! [X: A] :
              ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
             => aa(A,$o,aa(A,fun(A,$o),P,X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,abs_abs(A,Xd)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2592_arctan__double,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,Xd)) = aa(real,real,arctan,divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xd),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_2593_ceiling__log__eq__powr__iff,axiom,
    ! [Xd: real,B2: real,K: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),Xd)) = 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,$o,ord_less(real,powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),Xd)
            & aa(real,$o,ord_less_eq(real,Xd),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_2594_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(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,aa(real,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,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$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_2595_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT)] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summary))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ).

% height_compose_summary
tff(fact_2596_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(real,$o,ord_less_eq(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_2597_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Nb)
     => ( Deg = Nb ) ) ).

% deg_deg_n
tff(fact_2598_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)),TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList2,S2) ) ).

% deg_SUcn_Node
tff(fact_2599_VEBT_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: list(vEBT_VEBT),Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_2600_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_2601_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W: A,Z: A] :
          ( ( powr(A,W,Z) = zero_zero(A) )
        <=> ( W = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_2602_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_2603_zdvd1__eq,axiom,
    ! [Xd: int] :
      ( aa(int,$o,dvd_dvd(int,Xd),one_one(int))
    <=> ( abs_abs(int,Xd) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2604_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xd)) = Xd )
        <=> ? [N6: int] : Xd = aa(int,A,ring_1_of_int(A),N6) ) ) ).

% of_int_floor_cancel
tff(fact_2605_arctan__eq__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa(real,real,arctan,Xd) = zero_zero(real) )
    <=> ( Xd = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_2606_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_2607_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xd: A] :
          powr(A,Xd,zero_zero(A)) = $ite(Xd = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_2608_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_2609_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_2610_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less(int,abs_abs(int,Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2611_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_2612_powr__gt__zero,axiom,
    ! [Xd: real,A2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),powr(real,Xd,A2))
    <=> ( Xd != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_2613_powr__nonneg__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,powr(real,A2,Xd)),zero_zero(real))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_2614_powr__less__cancel__iff,axiom,
    ! [Xd: real,A2: real,B2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => ( aa(real,$o,ord_less(real,powr(real,Xd,A2)),powr(real,Xd,B2))
      <=> aa(real,$o,ord_less(real,A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_2615_arctan__less__zero__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,arctan,Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xd),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_2616_zero__less__arctan__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,arctan,Xd))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xd) ) ).

% zero_less_arctan_iff
tff(fact_2617_arctan__le__zero__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,arctan,Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xd),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_2618_zero__le__arctan__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arctan,Xd))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd) ) ).

% zero_le_arctan_iff
tff(fact_2619_powr__eq__one__iff,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( ( powr(real,A2,Xd) = one_one(real) )
      <=> ( Xd = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_2620_powr__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,one_one(real)) = Xd ) ) ).

% powr_one
tff(fact_2621_powr__one__gt__zero__iff,axiom,
    ! [Xd: real] :
      ( ( powr(real,Xd,one_one(real)) = Xd )
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd) ) ).

% powr_one_gt_zero_iff
tff(fact_2622_powr__le__cancel__iff,axiom,
    ! [Xd: real,A2: real,B2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,powr(real,Xd,A2)),powr(real,Xd,B2))
      <=> aa(real,$o,ord_less_eq(real,A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_2623_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_2624_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd) ) ) ).

% zero_le_floor
tff(fact_2625_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),V)),Xd) ) ) ).

% numeral_le_floor
tff(fact_2626_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xd)),zero_zero(int))
        <=> aa(A,$o,ord_less(A,Xd),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_2627_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xd)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less(A,Xd),aa(num,A,numeral_numeral(A),V)) ) ) ).

% floor_less_numeral
tff(fact_2628_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,one_one(A)),Xd) ) ) ).

% zero_less_floor
tff(fact_2629_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xd)),zero_zero(int))
        <=> aa(A,$o,ord_less(A,Xd),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_2630_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,one_one(int)),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,one_one(A)),Xd) ) ) ).

% one_le_floor
tff(fact_2631_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xd)),one_one(int))
        <=> aa(A,$o,ord_less(A,Xd),one_one(A)) ) ) ).

% floor_less_one
tff(fact_2632_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_2633_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xd),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xd)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_2634_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: num,Nb: nat] : archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xd)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb) ) ).

% floor_numeral_power
tff(fact_2635_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xd),one_one(A))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xd)),one_one(int)) ) ).

% floor_diff_one
tff(fact_2636_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_2637_powr__log__cancel,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
         => ( powr(real,A2,aa(real,real,log(A2),Xd)) = Xd ) ) ) ) ).

% powr_log_cancel
tff(fact_2638_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2))) = divide_divide(int,aa(num,int,numeral_numeral(int),A2),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_2639_powr__numeral,axiom,
    ! [Xd: real,Nb: num] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,aa(num,real,numeral_numeral(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),Nb)) ) ) ).

% powr_numeral
tff(fact_2640_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),Xd) ) ) ).

% numeral_less_floor
tff(fact_2641_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xd)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less(A,Xd),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_2642_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less(int,one_one(int)),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),Xd) ) ) ).

% one_less_floor
tff(fact_2643_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xd)),one_one(int))
        <=> aa(A,$o,ord_less(A,Xd),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_2644_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xd) ) ) ).

% neg_numeral_le_floor
tff(fact_2645_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xd)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less(A,Xd),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_neg_numeral
tff(fact_2646_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),B2))) = divide_divide(int,one_one(int),aa(num,int,numeral_numeral(int),B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_2647_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2)))) = 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_2648_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xd: A] :
          ( aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xd) ) ) ).

% neg_numeral_less_floor
tff(fact_2649_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xd)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less(A,Xd),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_2650_square__powr__half,axiom,
    ! [Xd: real] : powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))) = abs_abs(real,Xd) ).

% square_powr_half
tff(fact_2651_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),B2)))) = 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_2652_powr__powr,axiom,
    ! [Xd: real,A2: real,B2: real] : powr(real,powr(real,Xd,A2),B2) = powr(real,Xd,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)) ).

% powr_powr
tff(fact_2653_arctan__monotone,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xd),Y)
     => aa(real,$o,ord_less(real,aa(real,real,arctan,Xd)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone
tff(fact_2654_arctan__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,arctan,Xd)),aa(real,real,arctan,Y))
    <=> aa(real,$o,ord_less(real,Xd),Y) ) ).

% arctan_less_iff
tff(fact_2655_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xd)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,ord_less(A,Xd),Y) ) ) ).

% floor_less_cancel
tff(fact_2656_powr__non__neg,axiom,
    ! [A2: real,Xd: real] : ~ aa(real,$o,ord_less(real,powr(real,A2,Xd)),zero_zero(real)) ).

% powr_non_neg
tff(fact_2657_powr__less__mono2__neg,axiom,
    ! [A2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,A2),zero_zero(real))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,Xd),Y)
         => aa(real,$o,ord_less(real,powr(real,Y,A2)),powr(real,Xd,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_2658_powr__ge__pzero,axiom,
    ! [Xd: real,Y: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),powr(real,Xd,Y)) ).

% powr_ge_pzero
tff(fact_2659_powr__mono2,axiom,
    ! [A2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),Y)
         => aa(real,$o,ord_less_eq(real,powr(real,Xd,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mono2
tff(fact_2660_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( aa(real,$o,ord_less(real,one_one(real)),Xd)
       => aa(real,$o,ord_less(real,powr(real,Xd,A2)),powr(real,Xd,B2)) ) ) ).

% powr_less_mono
tff(fact_2661_powr__less__cancel,axiom,
    ! [Xd: real,A2: real,B2: real] :
      ( aa(real,$o,ord_less(real,powr(real,Xd,A2)),powr(real,Xd,B2))
     => ( aa(real,$o,ord_less(real,one_one(real)),Xd)
       => aa(real,$o,ord_less(real,A2),B2) ) ) ).

% powr_less_cancel
tff(fact_2662_powr__mono,axiom,
    ! [A2: real,B2: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,A2),B2)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
       => aa(real,$o,ord_less_eq(real,powr(real,Xd,A2)),powr(real,Xd,B2)) ) ) ).

% powr_mono
tff(fact_2663_abs__zmult__eq__1,axiom,
    ! [M: int,Nb: int] :
      ( ( abs_abs(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb)) = one_one(int) )
     => ( abs_abs(int,M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2664_abs__div,axiom,
    ! [Y: int,Xd: int] :
      ( aa(int,$o,dvd_dvd(int,Y),Xd)
     => ( abs_abs(int,divide_divide(int,Xd,Y)) = divide_divide(int,abs_abs(int,Xd),abs_abs(int,Y)) ) ) ).

% abs_div
tff(fact_2665_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_2666_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Z: int] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xd)),Z)
        <=> aa(A,$o,ord_less(A,Xd),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_2667_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xd)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_2668_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xd: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,Xd)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),Xd)) ) ).

% int_add_floor
tff(fact_2669_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: A] : aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xd)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y))) ) ).

% le_floor_add
tff(fact_2670_powr__mono2_H,axiom,
    ! [A2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,A2),zero_zero(real))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),Y)
         => aa(real,$o,ord_less_eq(real,powr(real,Y,A2)),powr(real,Xd,A2)) ) ) ) ).

% powr_mono2'
tff(fact_2671_powr__less__mono2,axiom,
    ! [A2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,Xd),Y)
         => aa(real,$o,ord_less(real,powr(real,Xd,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_2672_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Nb: nat] :
          ( ( Xd = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xd)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),archim6421214686448440834_floor(A,Xd)),Nb) ) ) ) ).

% floor_power
tff(fact_2673_powr__inj,axiom,
    ! [A2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,Xd) = powr(real,A2,Y) )
        <=> ( Xd = Y ) ) ) ) ).

% powr_inj
tff(fact_2674_gr__one__powr,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less(real,one_one(real)),powr(real,Xd,Y)) ) ) ).

% gr_one_powr
tff(fact_2675_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : archim6421214686448440834_floor(A,divide_divide(A,aa(int,A,ring_1_of_int(A),K),aa(int,A,ring_1_of_int(A),L))) = divide_divide(int,K,L) ) ).

% floor_divide_of_int_eq
tff(fact_2676_powr__le1,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
         => aa(real,$o,ord_less_eq(real,powr(real,Xd,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_2677_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
     => ( aa(real,$o,ord_less_eq(real,A2),B2)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
         => ( aa(real,$o,ord_less_eq(real,Xd),Y)
           => aa(real,$o,ord_less_eq(real,powr(real,Xd,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_2678_ge__one__powr__ge__zero,axiom,
    ! [Xd: real,A2: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
       => aa(real,$o,ord_less_eq(real,one_one(real)),powr(real,Xd,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_2679_powr__divide,axiom,
    ! [Xd: real,Y: real,A2: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( powr(real,divide_divide(real,Xd,Y),A2) = divide_divide(real,powr(real,Xd,A2),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_2680_powr__mult,axiom,
    ! [Xd: real,Y: real,A2: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xd,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_2681_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: 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_2682_log__base__powr,axiom,
    ! [A2: real,B2: real,Xd: real] :
      ( ( A2 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,A2,B2)),Xd) = divide_divide(real,aa(real,real,log(A2),Xd),B2) ) ) ).

% log_base_powr
tff(fact_2683_log__powr,axiom,
    ! [Xd: real,B2: real,Y: real] :
      ( ( Xd != zero_zero(real) )
     => ( aa(real,real,log(B2),powr(real,Xd,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(B2),Xd)) ) ) ).

% log_powr
tff(fact_2684_ln__powr,axiom,
    ! [Xd: real,Y: real] :
      ( ( Xd != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,Xd,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,ln_ln(real),Xd)) ) ) ).

% ln_powr
tff(fact_2685_zabs__def,axiom,
    ! [I: int] :
      abs_abs(int,I) = $ite(aa(int,$o,ord_less(int,I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_2686_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xd: A,A2: A,B2: A] : powr(A,Xd,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,Xd,A2)),powr(A,Xd,B2)) ) ).

% powr_add
tff(fact_2687_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,ord_less(int,abs_abs(int,modulo_modulo(int,K,L))),abs_abs(int,L)) ) ).

% abs_mod_less
tff(fact_2688_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)) = divide_divide(A,powr(A,W,Z1),powr(A,W,Z22)) ) ).

% powr_diff
tff(fact_2689_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,D),I)
       => aa(int,$o,ord_less_eq(int,abs_abs(int,D)),abs_abs(int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_2690_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),S: vEBT_VEBT,Xd: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts,S),Xd) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_2691_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_2692_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),Va2: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va2),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_2693_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),Va2: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d2(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va2),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_2694_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,Va2: nat] : vEBT_T_s_u_c_c(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_2695_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,Va2: nat] : vEBT_T_s_u_c_c2(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_2696_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xd)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),one_one(A))) ) ).

% one_add_floor
tff(fact_2697_floor__log__eq__powr__iff,axiom,
    ! [Xd: real,B2: real,K: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),Xd)) = K )
        <=> ( aa(real,$o,ord_less_eq(real,powr(real,B2,aa(int,real,ring_1_of_int(real),K))),Xd)
            & aa(real,$o,ord_less(real,Xd),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_2698_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,Nb: nat] : archim6421214686448440834_floor(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),divide_divide(nat,M,Nb)) ) ).

% floor_divide_of_nat_eq
tff(fact_2699_powr__realpow,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,aa(nat,real,semiring_1_of_nat(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb) ) ) ).

% powr_realpow
tff(fact_2700_less__log__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,Y),aa(real,real,log(B2),Xd))
        <=> aa(real,$o,ord_less(real,powr(real,B2,Y)),Xd) ) ) ) ).

% less_log_iff
tff(fact_2701_log__less__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(B2),Xd)),Y)
        <=> aa(real,$o,ord_less(real,Xd),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_2702_less__powr__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,Xd),powr(real,B2,Y))
        <=> aa(real,$o,ord_less(real,aa(real,real,log(B2),Xd)),Y) ) ) ) ).

% less_powr_iff
tff(fact_2703_powr__less__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less(real,powr(real,B2,Y)),Xd)
        <=> aa(real,$o,ord_less(real,Y),aa(real,real,log(B2),Xd)) ) ) ) ).

% powr_less_iff
tff(fact_2704_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          archimedean_ceiling(A,Xd) = $ite(Xd = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xd)),archim6421214686448440834_floor(A,Xd),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xd)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_2705_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,archimedean_ceiling(A,Xd)),archim6421214686448440834_floor(A,Xd))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_2706_floor__eq,axiom,
    ! [Nb: int,Xd: real] :
      ( aa(real,$o,ord_less(real,aa(int,real,ring_1_of_int(real),Nb)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),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,Xd) = Nb ) ) ) ).

% floor_eq
tff(fact_2707_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa(real,$o,ord_less(real,R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_2708_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa(real,$o,ord_less_eq(real,R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_2709_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_2710_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_2711_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)
      <=> ( abs_abs(int,Nb) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2712_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),S: vEBT_VEBT,Xd: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xd) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_2713_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,T))
        <=> ! [I2: int] :
              ( ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),I2)),T)
                & aa(A,$o,ord_less(A,T),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))) )
             => aa(int,$o,P,I2) ) ) ) ).

% floor_split
tff(fact_2714_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xd) = A2 )
        <=> ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A2)),Xd)
            & aa(A,$o,ord_less(A,Xd),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_2715_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xd: A] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),Xd)
         => ( aa(A,$o,ord_less(A,Xd),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,Xd) = Z ) ) ) ) ).

% floor_unique
tff(fact_2716_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xd: A,A2: A] : powr(A,Xd,aa(A,A,uminus_uminus(A),A2)) = divide_divide(A,one_one(A),powr(A,Xd,A2)) ) ).

% powr_minus_divide
tff(fact_2717_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xd: A] :
          ( aa(int,$o,ord_less(int,Z),archim6421214686448440834_floor(A,Xd))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xd) ) ) ).

% less_floor_iff
tff(fact_2718_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Z: int] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xd)),Z)
        <=> aa(A,$o,ord_less(A,Xd),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_2719_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B2)
           => aa(int,$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_2720_binomial__antimono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),K4)
     => ( aa(nat,$o,ord_less_eq(nat,divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,ord_less_eq(nat,K4),Nb)
         => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Nb),K4)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_2721_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_2722_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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_2723_binomial__mono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),K4)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),Nb)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K4)) ) ) ).

% binomial_mono
tff(fact_2724_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xd))),Xd)
          & aa(A,$o,ord_less(A,Xd),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xd)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_2725_powr__neg__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,aa(real,real,uminus_uminus(real),one_one(real))) = divide_divide(real,one_one(real),Xd) ) ) ).

% powr_neg_one
tff(fact_2726_powr__mult__base,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xd),powr(real,Xd,Y)) = powr(real,Xd,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_2727_le__log__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,log(B2),Xd))
        <=> aa(real,$o,ord_less_eq(real,powr(real,B2,Y)),Xd) ) ) ) ).

% le_log_iff
tff(fact_2728_log__le__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(B2),Xd)),Y)
        <=> aa(real,$o,ord_less_eq(real,Xd),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_2729_le__powr__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),powr(real,B2,Y))
        <=> aa(real,$o,ord_less_eq(real,aa(real,real,log(B2),Xd)),Y) ) ) ) ).

% le_powr_iff
tff(fact_2730_powr__le__iff,axiom,
    ! [B2: real,Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,$o,ord_less_eq(real,powr(real,B2,Y)),Xd)
        <=> aa(real,$o,ord_less_eq(real,Y),aa(real,real,log(B2),Xd)) ) ) ) ).

% powr_le_iff
tff(fact_2731_floor__eq2,axiom,
    ! [Nb: int,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(int,real,ring_1_of_int(real),Nb)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),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,Xd) = Nb ) ) ) ).

% floor_eq2
tff(fact_2732_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,divide_divide(real,A2,aa(int,real,ring_1_of_int(real),B2))) = divide_divide(int,archim6421214686448440834_floor(real,A2),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_2733_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,K)),L))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_abs_add_iff
tff(fact_2734_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),abs_abs(int,L)))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_add_abs_iff
tff(fact_2735_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P3,Q2)))),Q2)),P3) ) ) ).

% floor_divide_lower
tff(fact_2736_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,K),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$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_2737_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,Xd: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xd) = 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_2738_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,K),K4)
     => ( aa(nat,$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,$o,ord_less_eq(nat,K4),Nb)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(Nb),K4)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2739_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,K),K4)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),Nb)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K4)) ) ) ).

% binomial_strict_mono
tff(fact_2740_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,divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(Nb),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_2741_ln__powr__bound,axiom,
    ! [Xd: real,A2: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xd)),divide_divide(real,powr(real,Xd,A2),A2)) ) ) ).

% ln_powr_bound
tff(fact_2742_ln__powr__bound2,axiom,
    ! [Xd: real,A2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => aa(real,$o,ord_less_eq(real,powr(real,aa(real,real,ln_ln(real),Xd),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xd)) ) ) ).

% ln_powr_bound2
tff(fact_2743_log__add__eq__powr,axiom,
    ! [B2: real,Xd: 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)),Xd)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),Xd)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xd),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_2744_add__log__eq__powr,axiom,
    ! [B2: real,Xd: 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)),Xd)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),Xd)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),Xd)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_2745_minus__log__eq__powr,axiom,
    ! [B2: real,Xd: 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)),Xd)
         => ( aa(real,real,minus_minus(real,Y),aa(real,real,log(B2),Xd)) = aa(real,real,log(B2),divide_divide(real,powr(real,B2,Y),Xd)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_2746_nat__intermed__int__val,axiom,
    ! [M: nat,Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( aa(nat,$o,ord_less_eq(nat,M),I3)
            & aa(nat,$o,ord_less(nat,I3),Nb) )
         => aa(int,$o,ord_less_eq(int,abs_abs(int,aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(nat,$o,ord_less_eq(nat,M),Nb)
       => ( aa(int,$o,ord_less_eq(int,aa(nat,int,F2,M)),K)
         => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,F2,Nb))
           => ? [I3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M),I3)
                & aa(nat,$o,ord_less_eq(nat,I3),Nb)
                & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2747_incr__lemma,axiom,
    ! [D: int,Z: int,Xd: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => aa(int,$o,ord_less(int,Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,minus_minus(int,Xd),Z))),one_one(int))),D))) ) ).

% incr_lemma
tff(fact_2748_decr__lemma,axiom,
    ! [D: int,Xd: int,Z: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,Xd),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,minus_minus(int,Xd),Z))),one_one(int))),D))),Z) ) ).

% decr_lemma
tff(fact_2749_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,ord_less(A,P3),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,divide_divide(A,P3,Q2)))),one_one(A))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_2750_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary),Xd) = 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_2751_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : archimedean_round(A,Xd) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_2752_log__minus__eq__powr,axiom,
    ! [B2: real,Xd: 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)),Xd)
         => ( aa(real,real,minus_minus(real,aa(real,real,log(B2),Xd)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xd),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2753_nat__ivt__aux,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Nb)
         => aa(int,$o,ord_less_eq(int,abs_abs(int,aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,ord_less_eq(int,aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,F2,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I3),Nb)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2754_powr__neg__numeral,axiom,
    ! [Xd: real,Nb: num] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Nb))) = divide_divide(real,one_one(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),Nb))) ) ) ).

% powr_neg_numeral
tff(fact_2755_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Nb)
         => aa(int,$o,ord_less_eq(int,abs_abs(int,aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,ord_less_eq(int,aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,F2,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I3),Nb)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2756_arctan__add,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,$o,ord_less(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,Xd)),aa(real,real,arctan,Y)) = aa(real,real,arctan,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Y)))) ) ) ) ).

% arctan_add
tff(fact_2757_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archim6421214686448440834_floor(real,aa(real,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,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2758_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(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,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archim6421214686448440834_floor(real,aa(real,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_2759_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,ord_less_eq(nat,K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_2760_choose__two,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = 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_2761_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Nb)
     => ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeList,Summary)),Xd) ) ) ) ).

% both_member_options_ding
tff(fact_2762_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2763_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_2764_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_2765_intind,axiom,
    ! [A: $tType,I: nat,Nb: nat,P: fun(A,$o),Xd: A] :
      ( aa(nat,$o,ord_less(nat,I),Nb)
     => ( aa(A,$o,P,Xd)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Nb,Xd)),I)) ) ) ).

% intind
tff(fact_2766_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_2767_binomial__n__n,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),Nb) = one_one(nat) ).

% binomial_n_n
tff(fact_2768_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_2769_binomial__1,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ).

% binomial_1
tff(fact_2770_nth__replicate,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xd: A] :
      ( aa(nat,$o,ord_less(nat,I),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xd)),I) = Xd ) ) ).

% nth_replicate
tff(fact_2771_choose__one,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),one_one(nat)) = Nb ).

% choose_one
tff(fact_2772_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_2773_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_2774_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_2775_choose__mult__lemma,axiom,
    ! [M: nat,R2: 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),R2)),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),R2)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),M)) ).

% choose_mult_lemma
tff(fact_2776_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,R2),Nb)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Nb),R2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R2)) ) ).

% binomial_le_pow
tff(fact_2777_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(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_2778_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_2779_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = 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_2780_choose__mult,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$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_2781_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_2782_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_2783_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),Nb)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(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_2784_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa(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_2785_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_2786_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_2787_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_2788_in__children__def,axiom,
    ! [Nb: nat,TreeList: list(vEBT_VEBT),Xd: nat] :
      ( vEBT_V5917875025757280293ildren(Nb,TreeList,Xd)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xd,Nb))),vEBT_VEBT_low(Xd,Nb)) ) ).

% in_children_def
tff(fact_2789_heaphelp,axiom,
    ! [A: $tType,Xaa: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xba: vEBT_VEBTi,Nb: nat,Xca: vEBT_VEBTi,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,
          rep_assn(aa(assn,assn,
              aa(assn,fun(assn,assn),times_times(assn),
                aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,Xaa),Tree_is)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),Xba))),
                  pure_assn(( ( none(A) = none(A) )
                    & ( Nb = Nb ) )))),
              pure_assn(Xca = vEBT_Nodei(none(product_prod(nat,nat)),Nb,Xaa,Xba)))),
          H)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,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,TreeList,Summary)),Xca)),H) ) ).

% heaphelp
tff(fact_2790_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,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Uu) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_2791_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,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Uu) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_2792_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xd: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xd) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_2793_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Deg: nat,Summary: vEBT_VEBT] :
      ( member(vEBT_VEBT,T,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ) ).

% height_compose_child
tff(fact_2794_Rep__assn__inject,axiom,
    ! [Xd: assn,Y: assn] :
      ( ( rep_assn(Xd) = rep_assn(Y) )
    <=> ( Xd = Y ) ) ).

% Rep_assn_inject
tff(fact_2795_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),$o,finite_finite2(A),aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_2796_in__set__replicate,axiom,
    ! [A: $tType,Xd: A,Nb: nat,Y: A] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),replicate(A,Nb,Y)))
    <=> ( ( Xd = Y )
        & ( Nb != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_2797_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P,X5) )
    <=> ( aa(A,$o,P,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_2798_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P,X5) )
    <=> ( aa(A,$o,P,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_2799_mod__pure__star__dist,axiom,
    ! [P: assn,B2: $o,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2)))),H)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H)
        & (B2) ) ) ).

% mod_pure_star_dist
tff(fact_2800_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),pure_assn((B2))))
    <=> ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H2)
           => (B2) )
        & entails(P,Q) ) ) ).

% ent_pure_post_iff
tff(fact_2801_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: $o] :
      ( entails(P,pure_assn((B2)))
    <=> ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H2)
           => (B2) )
        & entails(P,one_one(assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_2802_mod__starE,axiom,
    ! [A2: assn,B2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),B2)),H)
     => ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(A2),X_1)
         => ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(B2),H_2) ) ) ).

% mod_starE
tff(fact_2803_mod__starD,axiom,
    ! [A3: assn,B4: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B4)),H)
     => ? [H1: product_prod(heap_ext(product_unit),set(nat)),H22: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(A3),H1)
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(B4),H22) ) ) ).

% mod_starD
tff(fact_2804_entails__def,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
    <=> ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H2)
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q),H2) ) ) ).

% entails_def
tff(fact_2805_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H3)
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q),H3) )
     => entails(P,Q) ) ).

% entailsI
tff(fact_2806_entailsD,axiom,
    ! [P: assn,Q: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( entails(P,Q)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H)
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q),H) ) ) ).

% entailsD
tff(fact_2807_ent__fwd,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat)),Q: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H)
     => ( entails(P,Q)
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q),H) ) ) ).

% ent_fwd
tff(fact_2808_finite__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A3 ) ).

% finite_list
tff(fact_2809_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xs)),B4)
    <=> ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
         => member(A,X5,B4) ) ) ).

% subset_code(1)
tff(fact_2810_list__assn__cong,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xs3: list(A),Xsi: list(B),Xsi2: list(B),A3: fun(A,fun(B,assn)),A6: fun(A,fun(B,assn))] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: A,Xi: B] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs3))
             => ( member(B,Xi,aa(list(B),set(B),set2(B),Xsi2))
               => ( aa(B,assn,aa(A,fun(B,assn),A3,X),Xi) = aa(B,assn,aa(A,fun(B,assn),A6,X),Xi) ) ) )
         => ( aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi) = aa(list(B),assn,vEBT_List_list_assn(A,B,A6,Xs3),Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_2811_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xd: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S),Xd) = vEBT_Node(Info,zero_zero(nat),Ts,S) ).

% vebt_insert.simps(2)
tff(fact_2812_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,ord_less_eq(nat,one_one(nat)),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_2813_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xd: A] : aa(A,A,arcosh(A),Xd) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),powr(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arcosh_def
tff(fact_2814_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          archimedean_round(A,Xd) = $ite(aa(A,$o,ord_less_eq(A,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xd)),archimedean_ceiling(A,Xd),archim6421214686448440834_floor(A,Xd)) ) ).

% round_altdef
tff(fact_2815_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xd: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xd) ).

% vebt_member.simps(2)
tff(fact_2816_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xd: A] : aa(A,A,arsinh(A),Xd) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arsinh_def
tff(fact_2817_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(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) )
         => ( ( M = aa(nat,nat,suc,Nb) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_2818_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Nb: nat] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
     => ( aa(nat,$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_2819_length__replicate,axiom,
    ! [A: $tType,Nb: nat,Xd: A] : aa(list(A),nat,size_size(list(A)),replicate(A,Nb,Xd)) = Nb ).

% length_replicate
tff(fact_2820_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xd: real] :
          ( ( real_Vector_of_real(A,Xd) = zero_zero(A) )
        <=> ( Xd = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_2821_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( real_Vector_of_real(A,zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_2822_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xd: real] :
          ( ( real_Vector_of_real(A,Xd) = one_one(A) )
        <=> ( Xd = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_2823_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_2824_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_2825_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xd: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,Xd)),real_Vector_of_real(A,Y)) ) ).

% of_real_mult
tff(fact_2826_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xd: real,Y: real] : real_Vector_of_real(A,divide_divide(real,Xd,Y)) = divide_divide(A,real_Vector_of_real(A,Xd),real_Vector_of_real(A,Y)) ) ).

% of_real_divide
tff(fact_2827_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xd: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xd)),real_Vector_of_real(A,Y)) ) ).

% of_real_add
tff(fact_2828_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xd: real,Nb: nat] : real_Vector_of_real(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),real_Vector_of_real(A,Xd)),Nb) ) ).

% of_real_power
tff(fact_2829_frac__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_frac(A,aa(int,A,ring_1_of_int(A),Z)) = zero_zero(A) ) ).

% frac_of_int
tff(fact_2830_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_2831_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xd)),one_one(A))) = abs_abs(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_2832_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xd)),aa(num,A,numeral_numeral(A),B2))) = abs_abs(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_2833_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_2834_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_2835_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xd: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xd) != aa(A,nat,size_size(A),Y) )
         => ( Xd != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_2836_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_2837_finite__maxlen,axiom,
    ! [A: $tType,M7: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M7)
     => ? [N: nat] :
        ! [X4: list(A)] :
          ( member(list(A),X4,M7)
         => aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),X4)),N) ) ) ).

% finite_maxlen
tff(fact_2838_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_2839_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),K)
         => ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P,I2),X_13) )
    <=> ? [Xs4: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs4) = K )
          & ! [I2: nat] :
              ( aa(nat,$o,ord_less(nat,I2),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I2),aa(nat,A,nth(A,Xs4),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_2840_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_2841_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,$o),P,Li),I3) )
     => ~ ! [L3: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L3) = Nb )
           => ~ ! [I4: nat] :
                  ( aa(nat,$o,ord_less(nat,I4),Nb)
                 => aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_2842_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xd: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y2: A] :
            ( member(A,Y2,aa(list(A),set(A),set2(A),Xs))
           => ( Y2 = Xd ) )
       => ( Xs = replicate(A,Nb,Xd) ) ) ) ).

% replicate_eqI
tff(fact_2843_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xd: A] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( X = Xd ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xd) = Xs ) ) ).

% replicate_length_same
tff(fact_2844_extract__pre__list__assn__lengthD,axiom,
    ! [B: $tType,A: $tType,A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,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_2845_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),archimedean_frac(A,Xd)) ) ).

% frac_ge_0
tff(fact_2846_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : aa(A,$o,ord_less(A,archimedean_frac(A,Xd)),one_one(A)) ) ).

% frac_lt_1
tff(fact_2847_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),one_one(A))) = archimedean_frac(A,Xd) ) ).

% frac_1_eq
tff(fact_2848_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,Xd: real] :
          ( ( Y != zero_zero(real) )
         => ( real_Vector_of_real(A,divide_divide(real,Xd,Y)) = divide_divide(A,real_Vector_of_real(A,Xd),real_Vector_of_real(A,Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_2849_length__pos__if__in__set,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_2850_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X5) )
    <=> ! [I2: nat] :
          ( aa(nat,$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_2851_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xd: 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,Xd,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xd) ) ) ).

% all_nth_imp_all_set
tff(fact_2852_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),L))
         => aa(A,$o,P,X5) )
    <=> ! [I2: nat] :
          ( aa(nat,$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_2853_in__set__conv__nth,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I2) = Xd ) ) ) ).

% in_set_conv_nth
tff(fact_2854_list__ball__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_2855_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_2856_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xd: A] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,Xd))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_2857_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( ( archimedean_frac(A,Xd) = Xd )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
            & aa(A,$o,ord_less(A,Xd),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_2858_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = $ite(aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xd)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xd)),archimedean_frac(A,Y)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xd)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_2859_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = $ite(aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xd)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xd)),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,Xd)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_2860_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(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) )
         => ( ( M = Nb )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_2861_mod__frame__fwd,axiom,
    ! [Ps: assn,H: product_prod(heap_ext(product_unit),set(nat)),P: assn,R: assn,F3: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Ps),H)
     => ( entails(P,R)
       => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F3))
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),F3)),H) ) ) ) ).

% mod_frame_fwd
tff(fact_2862_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ? [Y3: A] :
                ( aa(A,$o,P,Y3)
                & ~ aa(nat,$o,ord_less_eq(nat,aa(A,nat,F2,Y3)),aa(A,nat,F2,X)) ) )
       => ? [Y2: A] :
            ( aa(A,$o,P,Y2)
            & ~ aa(nat,$o,ord_less(nat,aa(A,nat,F2,Y2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_2863_log__base__10__eq1,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xd) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),aa(real,real,exp(real),one_one(real))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),aa(real,real,ln_ln(real),Xd)) ) ) ).

% log_base_10_eq1
tff(fact_2864_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_2865_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_2866_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,one_one(nat))
    <=> ? [A7: $o,B7: $o] : T = vEBT_Leaf((A7),(B7)) ) ).

% deg1Leaf
tff(fact_2867_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,one_one(nat))
     => ? [A5: $o,B3: $o] : T = vEBT_Leaf((A5),(B3)) ) ).

% deg_1_Leaf
tff(fact_2868_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A5: $o,B3: $o] : T = vEBT_Leaf((A5),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_2869_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y222)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y222) ) ) ) ).

% VEBT.inject(2)
tff(fact_2870_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_2871_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_2872_exp__less__cancel__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),Xd)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,ord_less(real,Xd),Y) ) ).

% exp_less_cancel_iff
tff(fact_2873_exp__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xd),Y)
     => aa(real,$o,ord_less(real,aa(real,real,exp(real),Xd)),aa(real,real,exp(real),Y)) ) ).

% exp_less_mono
tff(fact_2874_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_2875_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_2876_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_2877_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,exp(A),zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_2878_exp__eq__one__iff,axiom,
    ! [Xd: real] :
      ( ( aa(real,real,exp(real),Xd) = one_one(real) )
    <=> ( Xd = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_2879_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_2880_exp__less__one__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),Xd)),one_one(real))
    <=> aa(real,$o,ord_less(real,Xd),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_2881_one__less__exp__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,exp(real),Xd))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xd) ) ).

% one_less_exp_iff
tff(fact_2882_exp__le__one__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xd)),one_one(real))
    <=> aa(real,$o,ord_less_eq(real,Xd),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_2883_one__le__exp__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,exp(real),Xd))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd) ) ).

% one_le_exp_iff
tff(fact_2884_exp__ln__iff,axiom,
    ! [Xd: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xd)) = Xd )
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xd) ) ).

% exp_ln_iff
tff(fact_2885_exp__ln,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xd)) = Xd ) ) ).

% exp_ln
tff(fact_2886_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_2887_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_2888_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) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_2889_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_2890_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_2891_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,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_2892_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_2893_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_2894_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_2895_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_2896_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_2897_exp__less__cancel,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),Xd)),aa(real,real,exp(real),Y))
     => aa(real,$o,ord_less(real,Xd),Y) ) ).

% exp_less_cancel
tff(fact_2898_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_2899_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,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_2900_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_of_int
tff(fact_2901_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M)) ) ).

% take_bit_of_nat
tff(fact_2902_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : aa(A,A,exp(A),Xd) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_2903_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),aa(A,A,exp(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,exp(A),A3)) ) ).

% exp_times_arg_commute
tff(fact_2904_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_2905_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_2906_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv)),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_2907_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ? [X: real] : aa(real,real,exp(real),X) = Y ) ).

% exp_total
tff(fact_2908_exp__gt__zero,axiom,
    ! [Xd: real] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,exp(real),Xd)) ).

% exp_gt_zero
tff(fact_2909_not__exp__less__zero,axiom,
    ! [Xd: real] : ~ aa(real,$o,ord_less(real,aa(real,real,exp(real),Xd)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_2910_not__exp__le__zero,axiom,
    ! [Xd: real] : ~ aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xd)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_2911_exp__ge__zero,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,exp(real),Xd)) ).

% exp_ge_zero
tff(fact_2912_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_2913_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,ord_less_eq(nat,Nb),M),bit_se2584673776208193580ke_bit(A,Nb),bit_ri4674362597316999326ke_bit(A,M)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_2914_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xd) )
         => ( aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xd)),aa(A,A,exp(A),Y)) ) ) ) ).

% exp_add_commuting
tff(fact_2915_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xd)),aa(A,A,exp(A),Y)) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) ) ).

% mult_exp_exp
tff(fact_2916_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),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = $ite(aa(nat,$o,ord_less_eq(nat,Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_2917_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),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = $ite(aa(nat,$o,ord_less_eq(nat,Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_2918_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : aa(A,A,exp(A),aa(A,A,minus_minus(A,Xd),Y)) = divide_divide(A,aa(A,A,exp(A),Xd),aa(A,A,exp(A),Y)) ) ).

% exp_diff
tff(fact_2919_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,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_2920_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_2921_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_2922_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_2923_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$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_2924_exp__gt__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less(real,one_one(real)),aa(real,real,exp(real),Xd)) ) ).

% exp_gt_one
tff(fact_2925_exp__ge__add__one__self,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd)),aa(real,real,exp(real),Xd)) ).

% exp_ge_add_one_self
tff(fact_2926_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xd)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xd))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_2927_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Nb: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xd),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xd)),Nb) ) ).

% exp_of_nat2_mult
tff(fact_2928_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xd: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),Xd)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xd)),Nb) ) ).

% exp_of_nat_mult
tff(fact_2929_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_2930_log__ln,axiom,
    ! [Xd: real] : aa(real,real,ln_ln(real),Xd) = aa(real,real,log(aa(real,real,exp(real),one_one(real))),Xd) ).

% log_ln
tff(fact_2931_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xd: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),Xd)
    <=> $ite(
          Xd = zero_zero(nat),
          (A2),
          $ite(Xd = one_one(nat),(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_2932_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_2933_vebt__insert_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xd: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xd) = $ite(
        Xd = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(Xd = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_2934_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_2935_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_2936_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_2937_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_2938_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xd: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),Xd)
    <=> $ite(
          Xd = zero_zero(nat),
          (A2),
          $ite(Xd = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2939_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,Va2: nat] : vEBT_T_p_r_e_d2(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_2940_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_2941_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_2942_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_2943_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_2944_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_2945_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_2946_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_2947_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_2948_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_2949_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_2950_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,Q) ) ).

% is_entails
tff(fact_2951_exp__ge__add__one__self__aux,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd)),aa(real,real,exp(real),Xd)) ) ).

% exp_ge_add_one_self_aux
tff(fact_2952_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
          & aa(real,$o,ord_less_eq(real,X),aa(real,real,minus_minus(real,Y),one_one(real)))
          & ( aa(real,real,exp(real),X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_2953_ln__ge__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,ln_ln(real),Xd))
      <=> aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Y)),Xd) ) ) ).

% ln_ge_iff
tff(fact_2954_ln__x__over__x__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),Y)
       => aa(real,$o,ord_less_eq(real,divide_divide(real,aa(real,real,ln_ln(real),Y),Y)),divide_divide(real,aa(real,real,ln_ln(real),Xd),Xd)) ) ) ).

% ln_x_over_x_mono
tff(fact_2955_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xd: A,A2: A] :
          powr(A,Xd,A2) = $ite(Xd = zero_zero(A),zero_zero(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),Xd)))) ) ).

% powr_def
tff(fact_2956_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_2957_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_2958_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_2959_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_2960_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,Va2: nat] :
      vEBT_T_p_r_e_d(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = 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_2961_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,Xd: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A2),(B2)),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(Xd = 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_2962_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_2963_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_2964_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_2965_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = 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_2966_exp__le,axiom,
    aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_2967_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xd: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),divide_divide(A,Xd,aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = aa(A,A,exp(A),Xd) ) ) ) ).

% exp_divide_power_eq
tff(fact_2968_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(A,A,tanh(A),Xd) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,exp(A),Xd)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xd))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Xd)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xd)))) ) ).

% tanh_altdef
tff(fact_2969_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_2970_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_2971_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_2972_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_2973_exp__half__le2,axiom,
    aa(real,$o,ord_less_eq(real,aa(real,real,exp(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_2974_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_2975_num_Osize_I5_J,axiom,
    ! [X2: num] : aa(num,nat,size_size(num),bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_2976_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_2977_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y2: A] :
            ( aa(A,$o,P,Y2)
           => aa(nat,$o,ord_less(nat,aa(A,nat,F2,Y2)),B2) )
       => ? [X: A] :
            ( aa(A,$o,P,X)
            & ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(nat,$o,ord_less_eq(nat,aa(A,nat,F2,Y3)),aa(A,nat,F2,X)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_2978_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_2979_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,Xd: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A2),(B2)),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(Xd = 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_2980_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_2981_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_2982_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_2983_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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_2984_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F3: assn,Q: assn] :
      ( entails(P,R)
     => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F3))
       => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),F3),Q)
         => entails(Ps,Q) ) ) ) ).

% ent_frame_fwd
tff(fact_2985_fr__rot__rhs,axiom,
    ! [A3: assn,B4: assn,C6: assn] :
      ( entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),C6))
     => entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C6),B4)) ) ).

% fr_rot_rhs
tff(fact_2986_fr__refl,axiom,
    ! [A3: assn,B4: assn,C6: assn] :
      ( entails(A3,B4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C6),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),C6)) ) ).

% fr_refl
tff(fact_2987_fr__rot,axiom,
    ! [A3: assn,B4: assn,C6: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B4),C6)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),A3),C6) ) ).

% fr_rot
tff(fact_2988_exp__bound,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xd)),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)),Xd)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_2989_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_2990_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_2991_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( 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_2992_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_2993_real__exp__bound__lemma,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xd)),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))),Xd))) ) ) ).

% real_exp_bound_lemma
tff(fact_2994_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Nb))),Xd)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,Xd,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),aa(real,real,exp(real),Xd)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2995_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less_eq(real,Xd),aa(nat,real,semiring_1_of_nat(real),Nb))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,one_one(real)),divide_divide(real,Xd,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),Xd))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2996_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_2997_exp__lower__Taylor__quadratic,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xd)),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,exp(real),Xd)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2998_log__base__10__eq2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xd) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),Xd)) ) ) ).

% log_base_10_eq2
tff(fact_2999_tanh__real__altdef,axiom,
    ! [Xd: real] : aa(real,real,tanh(real),Xd) = divide_divide(real,aa(real,real,minus_minus(real,one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),Xd)))) ).

% tanh_real_altdef
tff(fact_3000_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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_3001_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_3002_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_3003_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_3004_uint32_Osize__eq,axiom,
    ! [P3: uint32] : aa(uint32,nat,size_size(uint32),P3) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))) ).

% uint32.size_eq
tff(fact_3005_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,ord_less(nat,From),To)
     => ( aa(nat,$o,ord_less_eq(nat,To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,To),From))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).

% slice_nth
tff(fact_3006_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_3007_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_3008_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_3009_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_3010_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_3011_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_3012_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_3013_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Nb: nat,Q2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => aa(nat,$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_3014_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_3015_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_3016_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_3017_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_3018_num__induct,axiom,
    ! [P: fun(num,$o),Xd: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X: num] :
            ( aa(num,$o,P,X)
           => aa(num,$o,P,inc(X)) )
       => aa(num,$o,P,Xd) ) ) ).

% num_induct
tff(fact_3019_add__inc,axiom,
    ! [Xd: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),Y)) ).

% add_inc
tff(fact_3020_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Nb: nat,K: int] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => aa(int,$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_3021_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_nonnegative
tff(fact_3022_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_3023_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_3024_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less(int,K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_3025_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_3026_inc_Osimps_I3_J,axiom,
    ! [Xd: num] : inc(aa(num,num,bit1,Xd)) = bit0(inc(Xd)) ).

% inc.simps(3)
tff(fact_3027_inc_Osimps_I2_J,axiom,
    ! [Xd: num] : inc(bit0(Xd)) = aa(num,num,bit1,Xd) ).

% inc.simps(2)
tff(fact_3028_add__One,axiom,
    ! [Xd: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),one2) = inc(Xd) ).

% add_One
tff(fact_3029_mult__inc,axiom,
    ! [Xd: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),Xd),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),Xd),Y)),Xd) ).

% mult_inc
tff(fact_3030_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_3031_option_Osize__neq,axiom,
    ! [A: $tType,Xd: option(A)] : aa(option(A),nat,size_size(option(A)),Xd) != zero_zero(nat) ).

% option.size_neq
tff(fact_3032_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xd: num] : aa(num,A,numeral_numeral(A),inc(Xd)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xd)),one_one(A)) ) ).

% numeral_inc
tff(fact_3033_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_3034_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_3035_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_3036_take__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M) = 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_3037_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_3038_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_3039_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = 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_3040_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_3041_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,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_3042_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_3043_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,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_3044_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(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_3045_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        & aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_3046_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_3047_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_3048_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_3049_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(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_3050_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_3051_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_3052_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),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_3053_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W))
          & aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_range_size
tff(fact_3054_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_3055_powr__int,axiom,
    ! [Xd: real,I: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),nat2(I)),divide_divide(real,one_one(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),nat2(aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_3056_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,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_3057_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( aa(nat,$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_3058_set__decode__inverse,axiom,
    ! [Nb: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(Nb)) = Nb ).

% set_decode_inverse
tff(fact_3059_unsigned__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),zero_zero(word(B))) = zero_zero(A) ) ) ).

% unsigned_0
tff(fact_3060_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_3061_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3062_nat__numeral,axiom,
    ! [K: num] : nat2(aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_3063_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3064_signed__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( ring_1_signed(B,A,zero_zero(word(B))) = zero_zero(A) ) ) ).

% signed_0
tff(fact_3065_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_nonnegative
tff(fact_3066_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),zero_zero(int))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),Xd) = zero_zero(int) ) ) ) ).

% uint_le_0_iff
tff(fact_3067_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xd)) ) ).

% uint_ge_0
tff(fact_3068_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : ~ aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),zero_zero(int)) ) ).

% uint_lt_0
tff(fact_3069_More__Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( ring_1_signed(A,int,Xd) = zero_zero(int) )
        <=> ( Xd = zero_zero(word(A)) ) ) ) ).

% More_Word.sint_0
tff(fact_3070_set__encode__inverse,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_3071_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_3072_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_3073_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_3074_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,Z),zero_zero(int))
     => ( nat2(Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_3075_nat__0__iff,axiom,
    ! [I: int] :
      ( ( nat2(I) = zero_zero(nat) )
    <=> aa(int,$o,ord_less_eq(int,I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_3076_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_3077_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_3078_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_3079_nat__zminus__int,axiom,
    ! [Nb: nat] : nat2(aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_3080_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_3081_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_3082_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_3083_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_3084_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
         => ( aa(nat,A,semiring_1_of_nat(A),nat2(Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_3085_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V3)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_3086_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_3087_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xd)),Nb) = nat2(Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_3088_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( nat2(Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xd)),Nb) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_3089_sint__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( ring_1_signed(A,int,Xd) = aa(int,int,uminus_uminus(int),one_one(int)) )
        <=> ( Xd = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% sint_minus1
tff(fact_3090_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(nat,$o,dvd_dvd(nat,Nb),nat2(abs_abs(int,K)))
    <=> aa(int,$o,dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ).

% dvd_nat_abs_iff
tff(fact_3091_nat__abs__dvd__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,nat2(abs_abs(int,K))),Nb)
    <=> aa(int,$o,dvd_dvd(int,K),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_3092_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_3093_word__mod__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_3094_nat__ceiling__le__eq,axiom,
    ! [Xd: real,A2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(archimedean_ceiling(real,Xd))),A2)
    <=> aa(real,$o,ord_less_eq(real,Xd),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_3095_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_3096_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_3097_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A2: int] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xd)),Nb)),nat2(A2))
    <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_3098_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xd: num,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xd)),Nb))
    <=> aa(int,$o,ord_less(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_3099_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A2: int] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xd)),Nb)),nat2(A2))
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_3100_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xd: num,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xd)),Nb))
    <=> aa(int,$o,ord_less_eq(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xd)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_3101_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_3102_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: nat,Xd: word(A)] :
          ( ( nat2(aa(word(A),int,semiring_1_unsigned(A,int),Y)) = Z )
         => ( aa(word(A),$o,ord_less(word(A),Xd),Y)
           => aa(nat,$o,ord_less(nat,nat2(aa(word(A),int,semiring_1_unsigned(A,int),Xd))),Z) ) ) ) ).

% nat_uint_less_helper
tff(fact_3103_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_3104_nat__numeral__as__int,axiom,
    ! [X4: num] : aa(num,nat,numeral_numeral(nat),X4) = nat2(aa(num,int,numeral_numeral(int),X4)) ).

% nat_numeral_as_int
tff(fact_3105_nat__zero__as__int,axiom,
    zero_zero(nat) = nat2(zero_zero(int)) ).

% nat_zero_as_int
tff(fact_3106_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_zero
tff(fact_3107_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_3108_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_3109_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,ord_less_eq(A,one_one(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_1
tff(fact_3110_nat__one__as__int,axiom,
    one_one(nat) = nat2(one_one(int)) ).

% nat_one_as_int
tff(fact_3111_Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,zero_zero(word(A))) = zero_zero(int) ) ) ).

% Word.sint_0
tff(fact_3112_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ? [X5: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X5)
          & aa(nat,$o,P,nat2(X5)) ) ) ).

% ex_nat
tff(fact_3113_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ! [X5: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X5)
         => aa(nat,$o,P,nat2(X5)) ) ) ).

% all_nat
tff(fact_3114_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z4)
       => ( ( nat2(Z) = nat2(Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_3115_uint__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = zero_zero(int) ) ) ).

% uint_0_eq
tff(fact_3116_uint__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),Xd) = zero_zero(int) )
        <=> ( Xd = zero_zero(word(A)) ) ) ) ).

% uint_0_iff
tff(fact_3117_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$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_3118_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_3119_word__less__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),B2)
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) ) ).

% word_less_def
tff(fact_3120_set__encode__eq,axiom,
    ! [A3: set(nat),B4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( aa(set(nat),$o,finite_finite2(nat),B4)
       => ( ( aa(set(nat),nat,nat_set_encode,A3) = aa(set(nat),nat,nat_set_encode,B4) )
        <=> ( A3 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_3121_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_greater_eq
tff(fact_3122_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_3123_uint__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),divide_divide(word(A),V,W)) = 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_3124_uint__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),divide_divide(word(A),Xd,Y)) = divide_divide(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_div
tff(fact_3125_uint__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),modulo_modulo(word(A),Xd,Y)) = modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_mod
tff(fact_3126_uint__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),modulo_modulo(word(A),V,W)) = modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),V),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_mod_distrib
tff(fact_3127_unsigned__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [W: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% unsigned_eq_0_iff
tff(fact_3128_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_3129_unset__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),Nb) = nat2(aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% unset_bit_nat_def
tff(fact_3130_signed__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [W: word(B)] :
          ( ( ring_1_signed(B,A,W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% signed_eq_0_iff
tff(fact_3131_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa(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_3132_dvd__fact,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),M)
     => ( aa(nat,$o,ord_less_eq(nat,M),Nb)
       => aa(nat,$o,dvd_dvd(nat,M),semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_3133_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_3134_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_3135_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_3136_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) ) ).

% nat_0_le
tff(fact_3137_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = nat2(Z) )
        & aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_3138_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_3139_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_3140_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ( modulo_modulo(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3141_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_3142_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xaa: word(A),Xd: word(B)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xaa)),aa(word(B),int,semiring_1_unsigned(B,int),Xd))) ) ).

% uint_add_ge0
tff(fact_3143_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xaa: word(A),Xd: word(B)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xaa)),aa(word(B),int,semiring_1_unsigned(B,int),Xd))) ) ).

% uint_mult_ge0
tff(fact_3144_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$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_3145_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : nat2(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(abs_abs(int,W))),nat2(abs_abs(int,Z))) ).

% nat_abs_mult_distrib
tff(fact_3146_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : divide_divide(word(A),A2,B2) = aa(int,word(A),ring_1_of_int(word(A)),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_3147_word__mod__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : modulo_modulo(word(A),A2,B2) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),A2),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_mod_def
tff(fact_3148_real__nat__ceiling__ge,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,Xd),aa(nat,real,semiring_1_of_nat(real),nat2(archimedean_ceiling(real,Xd)))) ).

% real_nat_ceiling_ge
tff(fact_3149_set__encode__inf,axiom,
    ! [A3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),A3)
     => ( aa(set(nat),nat,nat_set_encode,A3) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_3150_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_3151_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,R2),Nb)
     => aa(nat,$o,ord_less_eq(nat,divide_divide(nat,semiring_char_0_fact(nat,Nb),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Nb),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_3152_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),R2)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),nat2(archim6421214686448440834_floor(A,R2)))),R2) ) ) ).

% of_nat_floor
tff(fact_3153_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(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_3154_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),W)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less(int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_3155_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( nat2(W) = M )
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_3156_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = nat2(W) )
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_3157_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,nat2(I))
    <=> ( ! [N6: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N6) )
           => aa(nat,$o,P,N6) )
        & ( aa(int,$o,ord_less(int,I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_3158_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,ord_less(int,zero_zero(int)),W)
        | aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) )
     => ( aa(nat,$o,ord_less_eq(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less_eq(int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_3159_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z4)
       => ( nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(Z)),nat2(Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_3160_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(nat,$o,ord_less_eq(nat,Nb),nat2(K))
      <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ) ).

% le_nat_iff
tff(fact_3161_Suc__as__int,axiom,
    ! [X4: nat] : aa(nat,nat,suc,X4) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),one_one(int))) ).

% Suc_as_int
tff(fact_3162_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(nat,$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_3163_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(Z)),nat2(Z4)) ) ) ).

% nat_mult_distrib
tff(fact_3164_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z4)
     => ( aa(int,$o,ord_less_eq(int,Z4),Z)
       => ( nat2(aa(int,int,minus_minus(int,Z),Z4)) = aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_3165_nat__diff__distrib_H,axiom,
    ! [Xd: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => ( nat2(aa(int,int,minus_minus(int,Xd),Y)) = aa(nat,nat,minus_minus(nat,nat2(Xd)),nat2(Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_3166_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,ord_less_eq(nat,nat2(abs_abs(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_3167_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_3168_nat__div__distrib_H,axiom,
    ! [Y: int,Xd: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( nat2(divide_divide(int,Xd,Y)) = divide_divide(nat,nat2(Xd),nat2(Y)) ) ) ).

% nat_div_distrib'
tff(fact_3169_nat__div__distrib,axiom,
    ! [Xd: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => ( nat2(divide_divide(int,Xd,Y)) = divide_divide(nat,nat2(Xd),nat2(Y)) ) ) ).

% nat_div_distrib
tff(fact_3170_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( nat2(aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),nat2(Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_3171_nat__mod__distrib,axiom,
    ! [Xd: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => ( nat2(modulo_modulo(int,Xd,Y)) = modulo_modulo(nat,nat2(Xd),nat2(Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_3172_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$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_3173_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,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_3174_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,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_3175_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
         => ( aa(int,word(A),ring_1_of_int(word(A)),Xd) = aa(nat,word(A),semiring_1_of_nat(word(A)),nat2(Xd)) ) ) ) ).

% word_of_int_nat
tff(fact_3176_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : divide_divide(int,abs_abs(int,K),abs_abs(int,L)) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L)))) ).

% div_abs_eq_div_nat
tff(fact_3177_nat__floor__neg,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
     => ( nat2(archim6421214686448440834_floor(real,Xd)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_3178_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : modulo_modulo(int,abs_abs(int,K),abs_abs(int,L)) = aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_3179_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q2: word(A),Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$o,ord_less(word(A),P3),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = 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,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K6)),Q2) ) ) ) ) ).

% udvd_incr0
tff(fact_3180_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_3181_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(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_3182_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(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_3183_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q2: word(A),Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$o,ord_less(word(A),P3),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = 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,ord_less_eq(word(A),P3),aa(word(A),word(A),minus_minus(word(A),Q2),K6)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_3184_floor__eq3,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Nb)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( nat2(archim6421214686448440834_floor(real,Xd)) = Nb ) ) ) ).

% floor_eq3
tff(fact_3185_le__nat__floor,axiom,
    ! [Xd: nat,A2: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),Xd)),A2)
     => aa(nat,$o,ord_less_eq(nat,Xd),nat2(archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_3186_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_3187_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = 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_3188_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(nat,nat,suc,nat2(Z)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_3189_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),W)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),M)
      <=> aa(int,$o,ord_less(int,W),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_3190_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,Z),zero_zero(int))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = 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),Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_3191_floor__eq4,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),Nb)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( nat2(archim6421214686448440834_floor(real,Xd)) = Nb ) ) ) ).

% floor_eq4
tff(fact_3192_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q2: word(A),Ua: int,Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$o,ord_less(word(A),P3),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = 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,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K6)),Q2) ) ) ) ) ).

% udvd_incr'
tff(fact_3193_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q2: word(A),Ua: int,Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$o,ord_less(word(A),P3),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = 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,ord_less_eq(word(A),P3),aa(word(A),word(A),minus_minus(word(A),Q2),K6)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_3194_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa(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_3195_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,ord_less(int,K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),nat2(K))) ) ).

% of_int_of_nat
tff(fact_3196_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,nat2(Z)),M)
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Z),aa(int,$o,dvd_dvd(int,Z),aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_3197_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_3198_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_3199_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_3200_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$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))) = 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_3201_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = 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_3202_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),nat2(K))
      <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_3203_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W))
          & aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),W))) ) ) ).

% uint_range_size
tff(fact_3204_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_3205_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xd: int] :
          ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))),Xd)
         => aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),Xd) ) ) ).

% sint_above_size
tff(fact_3206_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int,W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,Xd),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))))
         => aa(int,$o,ord_less_eq(int,Xd),ring_1_signed(A,int,W)) ) ) ).

% sint_below_size
tff(fact_3207_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),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),Xd)),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)),Xd))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_3208_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)),Xd),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),Xd)),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)),Xd))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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),Xd)),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)),Xd)))) ) ).

% uint_plus_if_size
tff(fact_3209_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xd),Y)) = $ite(aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xd)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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),Xd)),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)),Xd)))) ) ).

% uint_sub_if_size
tff(fact_3210_sin__coeff__def,axiom,
    ! [X4: nat] :
      sin_coeff(X4) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4),zero_zero(real),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,minus_minus(nat,X4),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),semiring_char_0_fact(real,X4))) ).

% sin_coeff_def
tff(fact_3211_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)),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_3212_cos__coeff__def,axiom,
    ! [X4: nat] :
      cos_coeff(X4) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,X4,aa(num,nat,numeral_numeral(nat),bit0(one2)))),semiring_char_0_fact(real,X4)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_3213_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_3214_divide__int__def,axiom,
    ! [K: int,L: int] :
      divide_divide(int,K,L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,dvd_dvd(int,L),K)))))) ) ).

% divide_int_def
tff(fact_3215_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% sgn_sgn
tff(fact_3216_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_3217_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_3218_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_3219_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_3220_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,divide_divide(A,A2,B2)) = divide_divide(A,sgn_sgn(A,A2),sgn_sgn(A,B2)) ) ).

% sgn_divide
tff(fact_3221_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),sgn_sgn(A,A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_3222_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : 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),sgn_sgn(A,A2)),Nb) ) ).

% power_sgn
tff(fact_3223_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,sgn_sgn(A,A2)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_3224_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),sgn_sgn(A,A2))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_3225_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,sgn_sgn(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),sgn_sgn(A,B2)) ) ).

% divide_sgn
tff(fact_3226_sgn__uint__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : sgn_sgn(int,aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa($o,int,zero_neq_one_of_bool(int),W != zero_zero(word(A))) ) ).

% sgn_uint_eq
tff(fact_3227_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3228_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3229_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_3230_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( abs_abs(A,sgn_sgn(A,A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_3231_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_3232_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : abs_abs(A,sgn_sgn(A,A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_3233_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,abs_abs(A,A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3234_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( Xd != 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),Xd),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xd) ) ) ) ).

% Suc_unat_minus_one
tff(fact_3235_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),K),sgn_sgn(int,R2)))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3236_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),K))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3237_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),sgn_sgn(int,R2))),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3238_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),L)),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3239_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_3240_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : sgn_sgn(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)) ) ).

% sgn_of_nat
tff(fact_3241_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_3242_ucast__0__I,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xd: word(A)] :
          ( ( Xd = zero_zero(word(A)) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xd) = zero_zero(word(B)) ) ) ) ).

% ucast_0_I
tff(fact_3243_ucast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% ucast_0
tff(fact_3244_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_3245_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_3246_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] :
          ( ( sgn_sgn(A,Xd) = zero_zero(A) )
        <=> ( Xd = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_3247_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: 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),sgn_sgn(A,A2)),sgn_sgn(A,B2)) ) ).

% sgn_mult
tff(fact_3248_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: A,Y: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xd)),sgn_sgn(A,Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_3249_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( sgn_sgn(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = sgn_sgn(A,A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_3250_word__unat__Rep__inject1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xd) = aa(word(B),nat,semiring_1_unsigned(B,nat),one_one(word(B))) )
        <=> ( Xd = one_one(word(A)) ) ) ) ).

% word_unat_Rep_inject1
tff(fact_3251_scast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(B,word(A),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% scast_0
tff(fact_3252_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) != sgn_sgn(A,A2) )
         => ( ( sgn_sgn(A,A2) != zero_zero(A) )
           => ( ( sgn_sgn(A,B2) != zero_zero(A) )
             => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),sgn_sgn(A,B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_3253_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(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_3254_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( 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),abs_abs(A,A2)),abs_abs(A,B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_3255_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : abs_abs(A,K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),sgn_sgn(A,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_3256_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A2)),sgn_sgn(A,A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_3257_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),abs_abs(A,A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_3258_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xd)),abs_abs(A,Xd)) = Xd ) ).

% mult_sgn_abs
tff(fact_3259_unat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = zero_zero(nat) ) ) ).

% unat_0
tff(fact_3260_unat__eq__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xd) = zero_zero(nat) )
        <=> ( Xd = zero_zero(word(A)) ) ) ) ).

% unat_eq_zero
tff(fact_3261_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_3262_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_3263_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_3264_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
     => ( divide_divide(int,K,L) = divide_divide(int,abs_abs(int,K),abs_abs(int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_3265_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)),divide_divide(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Z),Nb))) = divide_divide(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Z),Nb) ) ).

% uno_simps(2)
tff(fact_3266_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_3267_unat__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),divide_divide(word(A),V,W)) = 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_3268_unat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),divide_divide(word(A),Xd,Y)) = divide_divide(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_div
tff(fact_3269_unat__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: nat] :
          ( aa(word(A),$o,ord_less(word(A),A2),B2)
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),B2) = C2 )
           => aa(word(A),$o,ord_less(word(A),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),C2)) ) ) ) ).

% unat_of_nat_less
tff(fact_3270_unat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),modulo_modulo(word(A),Xd,Y)) = modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_mod
tff(fact_3271_unat__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),modulo_modulo(word(A),V,W)) = modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),V),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_mod_distrib
tff(fact_3272_sin__coeff__Suc,axiom,
    ! [Nb: nat] : sin_coeff(aa(nat,nat,suc,Nb)) = divide_divide(real,cos_coeff(Nb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ).

% sin_coeff_Suc
tff(fact_3273_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = one_one(A) )
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_3274_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          abs_abs(A,sgn_sgn(A,A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_3275_unat__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xd) = aa(nat,nat,suc,zero_zero(nat)) )
        <=> ( Xd = one_one(word(A)) ) ) ) ).

% unat_eq_1
tff(fact_3276_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ aa(int,$o,dvd_dvd(int,L),K)
       => ( sgn_sgn(int,modulo_modulo(int,K,L)) = sgn_sgn(int,L) ) ) ) ).

% sgn_mod
tff(fact_3277_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd))
        <=> ( Xd != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_3278_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),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)),Xd),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_plus_simple
tff(fact_3279_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd))
         => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)),Xd) ) ) ).

% word_of_nat_less
tff(fact_3280_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xd),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),Xd)),Nb) ) ) ).

% unat_less_helper
tff(fact_3281_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_3282_cos__coeff__Suc,axiom,
    ! [Nb: nat] : cos_coeff(aa(nat,nat,suc,Nb)) = 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_3283_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_3284_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : divide_divide(word(A),A2,B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),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_3285_word__arith__nat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : modulo_modulo(word(A),A2,B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_mod
tff(fact_3286_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] :
          sgn_sgn(A,Xd) = $ite(
            Xd = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,ord_less(A,zero_zero(A)),Xd),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_3287_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_3288_zsgn__def,axiom,
    ! [I: 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_3289_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] :
          real_V7770717601297561774m_norm(A,sgn_sgn(A,Xd)) = $ite(Xd = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_3290_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero(int) )
     => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),abs_abs(int,K)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),abs_abs(int,L))) = divide_divide(int,abs_abs(int,K),abs_abs(int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_3291_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),Xd)
        <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)) ) ) ).

% unat_1_0
tff(fact_3292_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_3293_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),K)
     => ( 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),sgn_sgn(int,K)),sgn_sgn(int,L))),divide_divide(int,abs_abs(int,K),abs_abs(int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_3294_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_3295_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_3296_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),Xd)
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xd),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xd) ) ) ) ).

% Suc_unat_diff_1
tff(fact_3297_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_3298_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)),modulo_modulo(nat,M,aa(word(A),nat,semiring_1_unsigned(A,nat),Z)))) = modulo_modulo(nat,M,aa(word(A),nat,semiring_1_unsigned(A,nat),Z)) ) ) ) ).

% uno_simps(1)
tff(fact_3299_measure__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A)] :
          ( ( P3 != zero_zero(word(A)) )
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),P3),one_one(word(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),P3)) ) ) ).

% measure_unat
tff(fact_3300_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc: A] :
      set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc) = $ite(aa(nat,$o,ord_less(nat,B2),A2),Acc,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A2),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3301_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xd: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xca: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xd,Xaa,Xba,Xca) = Y )
     => ( Y = $ite(aa(nat,$o,ord_less(nat,Xba),Xaa),Xca,set_fo6178422350223883121st_nat(A,Xd,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xba,aa(A,A,aa(nat,fun(A,A),Xd,Xaa),Xca))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_3302_word__overflow__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)),Xd),one_one(word(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),one_one(nat)) )
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow_unat
tff(fact_3303_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_3304_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
       => ( divide_divide(int,K,L) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),divide_divide(int,abs_abs(int,K),abs_abs(int,L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_3305_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,MaxBound: word(A),Xd: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,ord_less(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)))
          <=> aa(word(A),$o,ord_less_eq(word(A),Xd),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_3306_odd__word__imp__even__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),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)),Xd),one_one(word(A))))) ) ) ) ).

% odd_word_imp_even_next
tff(fact_3307_even__word__imp__odd__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),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)),Xd),one_one(word(A))))) ) ) ) ).

% even_word_imp_odd_next
tff(fact_3308_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( ( divide_divide(word(A),Nb,M) = one_one(word(A)) )
        <=> ( aa(word(A),$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_3309_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 )
        <=> ? [Q5: 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),Q5),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_3310_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)),Xd),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),Xd)),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)),Xd))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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),Xd)),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)),Xd)))) ) ).

% unat_plus_if_size
tff(fact_3311_modulo__int__def,axiom,
    ! [K: int,L: int] :
      modulo_modulo(int,K,L) = $ite(
        L = zero_zero(int),
        K,
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L))))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,L)),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K)))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L))))))) ) ).

% modulo_int_def
tff(fact_3312_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xd),Y)) = $ite(aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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),Xd)),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)),Xd)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if_size
tff(fact_3313_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),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),Xd)),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)),Xd))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_3314_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(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),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M,Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,M,Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Nb),M)))))) ) ).

% divide_int_unfold
tff(fact_3315_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(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),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Nb),M))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,Nb))))) ) ).

% modulo_int_unfold
tff(fact_3316_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( divide_divide(int,A2,B2) != zero_zero(int) )
     => ( sgn_sgn(int,divide_divide(int,A2,B2)) = sgn_sgn(int,aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_3317_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A5: $o,B3: $o] : A1 = vEBT_Leaf((A5),(B3))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                     => vEBT_invar_vebt(X4,N) )
                 => ( vEBT_invar_vebt(Summary2,M2)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                     => ( ( M2 = N )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                         => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_12)
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                       => vEBT_invar_vebt(X4,N) )
                   => ( vEBT_invar_vebt(Summary2,M2)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                       => ( ( M2 = aa(nat,nat,suc,N) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                           => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_12)
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                         => vEBT_invar_vebt(X4,N) )
                     => ( vEBT_invar_vebt(Summary2,M2)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                         => ( ( M2 = N )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                             => ( ! [I4: nat] :
                                    ( aa(nat,$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_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                               => ( ( ( Mi = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
                                 => ( aa(nat,$o,ord_less_eq(nat,Mi),Ma2)
                                   => ( aa(nat,$o,ord_less(nat,Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                     => ~ ( ( Mi != Ma2 )
                                         => ! [I4: nat] :
                                              ( aa(nat,$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(Ma2,N) = I4 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                & ! [X4: nat] :
                                                    ( ( ( vEBT_VEBT_high(X4,N) = I4 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N)) )
                                                   => ( aa(nat,$o,ord_less(nat,Mi),X4)
                                                      & aa(nat,$o,ord_less_eq(nat,X4),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: 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),product_Pair(nat,nat,Mi),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_invar_vebt(X4,N) )
                       => ( vEBT_invar_vebt(Summary2,M2)
                         => ( ( 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))),M2) )
                           => ( ( M2 = aa(nat,nat,suc,N) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                               => ( ! [I4: nat] :
                                      ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                                 => ( ( ( Mi = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                         => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
                                   => ( aa(nat,$o,ord_less_eq(nat,Mi),Ma2)
                                     => ( aa(nat,$o,ord_less(nat,Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                       => ~ ( ( Mi != Ma2 )
                                           => ! [I4: nat] :
                                                ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                  & ! [X4: nat] :
                                                      ( ( ( vEBT_VEBT_high(X4,N) = I4 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N)) )
                                                     => ( aa(nat,$o,ord_less(nat,Mi),X4)
                                                        & aa(nat,$o,ord_less_eq(nat,X4),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3318_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A7: $o,B7: $o] : A1 = vEBT_Leaf((A7),(B7))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X5,N6) )
            & vEBT_invar_vebt(Summary3,N6)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),N6) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_13)
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
        | ? [TreeList3: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X5,N6) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N6))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),aa(nat,nat,suc,N6)) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_13)
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
        | ? [TreeList3: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X5,N6) )
            & vEBT_invar_vebt(Summary3,N6)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),N6) )
            & ! [I2: nat] :
                ( aa(nat,$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_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I2) ) )
            & ( ( Mi2 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
            & aa(nat,$o,ord_less_eq(nat,Mi2),Ma3)
            & aa(nat,$o,ord_less(nat,Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi2 != Ma3 )
             => ! [I2: nat] :
                  ( aa(nat,$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(Ma3,N6) = I2 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(Ma3,N6)) )
                    & ! [X5: nat] :
                        ( ( ( vEBT_VEBT_high(X5,N6) = I2 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(X5,N6)) )
                       => ( aa(nat,$o,ord_less(nat,Mi2),X5)
                          & aa(nat,$o,ord_less_eq(nat,X5),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X5,N6) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N6))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),aa(nat,nat,suc,N6)) )
            & ! [I2: nat] :
                ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I2) ) )
            & ( ( Mi2 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
            & aa(nat,$o,ord_less_eq(nat,Mi2),Ma3)
            & aa(nat,$o,ord_less(nat,Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi2 != Ma3 )
             => ! [I2: nat] :
                  ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N6) = I2 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(Ma3,N6)) )
                    & ! [X5: nat] :
                        ( ( ( vEBT_VEBT_high(X5,N6) = I2 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(X5,N6)) )
                       => ( aa(nat,$o,ord_less(nat,Mi2),X5)
                          & aa(nat,$o,ord_less_eq(nat,X5),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3319_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),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% and_int_unfold
tff(fact_3320_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.right_idem
tff(fact_3321_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : 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),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.left_idem
tff(fact_3322_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),A2) = A2 ) ).

% and.idem
tff(fact_3323_mi__eq__ma__no__ch,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi3 = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
             => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) )
          & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_3324_insert__simp__mima,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        | ( Xd = Ma ) )
     => ( aa(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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_3325_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_3326_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xd) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_3327_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_3328_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_3329_sgn__le__0__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,sgn_sgn(real,Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xd),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_3330_zero__le__sgn__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),sgn_sgn(real,Xd))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd) ) ).

% zero_le_sgn_iff
tff(fact_3331_not__None__eq,axiom,
    ! [A: $tType,Xd: option(A)] :
      ( ( Xd != none(A) )
    <=> ? [Y5: A] : Xd = aa(A,option(A),some(A),Y5) ) ).

% not_None_eq
tff(fact_3332_not__Some__eq,axiom,
    ! [A: $tType,Xd: option(A)] :
      ( ! [Y5: A] : Xd != aa(A,option(A),some(A),Y5)
    <=> ( Xd = none(A) ) ) ).

% not_Some_eq
tff(fact_3333_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_and
tff(fact_3334_tdeletemimi,axiom,
    ! [Deg: nat,Mi3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Mi3)),Deg,TreeList,Summary),Xd)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ).

% tdeletemimi
tff(fact_3335_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Mi3)),Deg,TreeList,Summary),Xd)),one_one(nat)) ) ).

% tdeletemimi'
tff(fact_3336_mi__ma__2__deg,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,Mi3),Ma)
        & aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_3337_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),aa(A,A,uminus_uminus(A),one_one(A))) = Xd ) ).

% bit.conj_one_right
tff(fact_3338_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_3339_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_3340_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
      ( ! [X5: A,Y5: B] : V != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X5),Y5))
    <=> ( V = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_3341_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        | aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_3342_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,ord_less(int,K),zero_zero(int))
        & aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_3343_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi3: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: nat] :
      ( aa(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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary)),Xd)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xd = Mi3 )
          | ( Xd = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3344_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A] : ~ aa(option(A),$o,ord_less_eq(option(A),aa(A,option(A),some(A),Xd)),none(A)) ) ).

% less_eq_option_Some_None
tff(fact_3345_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(option(A),$o,ord_less(option(A),aa(A,option(A),some(A),Xd)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,ord_less(A,Xd),Y) ) ) ).

% less_option_Some
tff(fact_3346_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A] : aa(option(A),$o,ord_less(option(A),none(A)),aa(A,option(A),some(A),Xd)) ) ).

% less_option_None_Some_code
tff(fact_3347_member__inv,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary)),Xd)
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
        & ( ( Xd = Mi3 )
          | ( Xd = Ma )
          | ( aa(nat,$o,ord_less(nat,Xd),Ma)
            & aa(nat,$o,ord_less(nat,Mi3),Xd)
            & aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_3348_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xd: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi3: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Deg)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary)),Xd) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3349_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xd))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_3350_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_3351_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_3352_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xd))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_3353_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_3354_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_3355_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_3356_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_3357_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: 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,Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_3358_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_3359_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_3360_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: 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,Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_3361_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_3362_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),C2)) = 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),B2),C2)) ) ).

% and.left_commute
tff(fact_3363_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),A2) ) ).

% and.commute
tff(fact_3364_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),C2) = 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),B2),C2)) ) ).

% and.assoc
tff(fact_3365_prod__decode__aux_Ocases,axiom,
    ! [Xd: product_prod(nat,nat)] :
      ~ ! [K2: nat,M2: nat] : Xd != aa(nat,product_prod(nat,nat),product_Pair(nat,nat,K2),M2) ).

% prod_decode_aux.cases
tff(fact_3366_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_and_eq
tff(fact_3367_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,Xd: option(A),P: fun(option(A),fun(option(B),$o)),Y: option(B)] :
      ( ( ( Xd = none(A) )
       => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xd),Y) )
     => ( ( ( Y = none(B) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xd),Y) )
       => ( ! [A5: A,B3: B] :
              ( ( Xd = aa(A,option(A),some(A),A5) )
             => ( ( Y = aa(B,option(B),some(B),B3) )
               => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xd),Y) ) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_3368_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ! [X_13: option(A)] : aa(option(A),$o,P,X_13)
    <=> ( aa(option(A),$o,P,none(A))
        & ! [X5: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X5)) ) ) ).

% split_option_all
tff(fact_3369_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ? [X_13: option(A)] : aa(option(A),$o,P,X_13)
    <=> ( aa(option(A),$o,P,none(A))
        | ? [X5: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X5)) ) ) ).

% split_option_ex
tff(fact_3370_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X22: A] : Y != aa(A,option(A),some(A),X22) ) ).

% option.exhaust
tff(fact_3371_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X2: A] :
      ( ( Option = aa(A,option(A),some(A),X2) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_3372_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X2: A] : none(A) != aa(A,option(A),some(A),X2) ).

% option.distinct(1)
tff(fact_3373_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xd: vEBT_VEBT] :
      ( ! [A5: $o,B3: $o] : Xd != vEBT_Leaf((A5),(B3))
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xd != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_3374_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),zero_zero(nat),Va2,Vb),Xd)
    <=> ( ( Xd = Mi3 )
        | ( Xd = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3375_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_3376_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xd: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less_eq(int,Y),Z)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xd),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_3377_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less_eq(int,Y),Z)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_3378_AND__upper2,axiom,
    ! [Y: int,Xd: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xd),Y)),Y) ) ).

% AND_upper2
tff(fact_3379_AND__upper1,axiom,
    ! [Xd: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xd),Y)),Xd) ) ).

% AND_upper1
tff(fact_3380_AND__lower,axiom,
    ! [Xd: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xd),Y)) ) ).

% AND_lower
tff(fact_3381_real__sgn__eq,axiom,
    ! [Xd: real] : sgn_sgn(real,Xd) = divide_divide(real,Xd,abs_abs(real,Xd)) ).

% real_sgn_eq
tff(fact_3382_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,
    ! [Mi3: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),zero_zero(nat),TreeList,Summary),Xd) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_3383_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,
    ! [Mi3: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),zero_zero(nat),TreeList,Summary),Xd) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_3384_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xd),Xd)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_3385_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,
    ! [Mi3: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),Xd) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_3386_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,
    ! [Mi3: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),Xd) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_3387_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: option(A)] :
          ( aa(option(A),$o,ord_less(option(A),none(A)),Xd)
         => ? [Z2: A] : Xd = aa(A,option(A),some(A),Z2) ) ) ).

% less_option_None_is_Some
tff(fact_3388_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A] : aa(option(A),$o,ord_less(option(A),none(A)),aa(A,option(A),some(A),Xd)) ) ).

% less_option_None_Some
tff(fact_3389_height__node,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Nb)
     => aa(nat,$o,ord_less_eq(nat,one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary))) ) ).

% height_node
tff(fact_3390_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xd: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xd),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_3391_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_3392_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_3393_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,
    ! [Xd: vEBT_VEBT] :
      ( ( Xd != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xd != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xd != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xd != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xd != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_3394_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xd: 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)),Xd) ).

% vebt_member.simps(3)
tff(fact_3395_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_3396_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_3397_sgn__real__def,axiom,
    ! [A2: 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_3398_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_3399_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc2: vEBT_VEBT,Xd: 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)),Xd) ).

% vebt_member.simps(4)
tff(fact_3400_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_3401_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_3402_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_3403_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_3404_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) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_3405_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)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_3406_sgn__power__injE,axiom,
    ! [A2: real,Nb: nat,Xd: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,A2)),Nb)) = Xd )
     => ( ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,B2)),Nb)) )
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_3407_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_3408_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_3409_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_3410_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_3411_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,Xd: 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),Xd) = 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_3412_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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% and_int_rec
tff(fact_3413_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,Xd: 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),Xd) = 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_3414_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(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) )
         => ( ( M = Nb )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I3) ) )
               => ( ( ( Mi3 = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa(nat,$o,ord_less_eq(nat,Mi3),Ma)
                   => ( aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi3 != Ma )
                         => ! [I3: nat] :
                              ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,Nb)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X,Nb)) )
                                   => ( aa(nat,$o,ord_less(nat,Mi3),X)
                                      & aa(nat,$o,ord_less_eq(nat,X),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_3415_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(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) )
         => ( ( M = aa(nat,nat,suc,Nb) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I3) ) )
               => ( ( ( Mi3 = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa(nat,$o,ord_less_eq(nat,Mi3),Ma)
                   => ( aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi3 != Ma )
                         => ! [I3: nat] :
                              ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,Nb)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X,Nb)) )
                                   => ( aa(nat,$o,ord_less(nat,Mi3),X)
                                      & aa(nat,$o,ord_less_eq(nat,X),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_3416_succ__list__to__short,axiom,
    ! [Deg: nat,Mi3: nat,Xd: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less_eq(nat,Mi3),Xd)
       => ( aa(nat,$o,ord_less_eq(nat,aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xd,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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_3417_pred__list__to__short,axiom,
    ! [Deg: nat,Xd: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi3: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less_eq(nat,Xd),Ma)
       => ( aa(nat,$o,ord_less_eq(nat,aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xd,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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_3418_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q2),R2)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),product_Pair(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),one_one(A))),aa(A,A,minus_minus(A,R2),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),product_Pair(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),R2)) ) ).

% divmod_step_eq
tff(fact_3419_geqmaxNone,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Nb: nat,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,Ma),Xd)
       => ( 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_3420_del__single__cont,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & ( Xd = Ma ) )
     => ( aa(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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ) ) ) ).

% del_single_cont
tff(fact_3421_power__shift,axiom,
    ! [Xd: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xd),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),Xd)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% power_shift
tff(fact_3422_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => vEBT_invar_vebt(vEBT_vebt_delete(T,Xd),Nb) ) ).

% delete_pres_valid
tff(fact_3423_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_delete(T,Xd)),Y)
      <=> ( ( Xd != Y )
          & aa(nat,$o,vEBT_V8194947554948674370ptions(T),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_3424_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_delete(T,Xd)),Y)
      <=> ( ( Xd != Y )
          & aa(nat,$o,vEBT_vebt_member(T),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_3425_pred__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Px: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T),Xd,Px) ) ) ).

% pred_corr
tff(fact_3426_succ__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T),Xd,Sx) ) ) ).

% succ_corr
tff(fact_3427_pred__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T),Xd,Sx) ) ) ).

% pred_correct
tff(fact_3428_succ__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T),Xd,Sx) ) ) ).

% succ_correct
tff(fact_3429_helpyd,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = 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_3430_helpypredd,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = 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_3431_word__bitwise__m1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)))),Xd) = Xd ) ).

% word_bitwise_m1_simps(2)
tff(fact_3432_word__and__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = Xd ) ).

% word_and_max
tff(fact_3433_delt__out__of__range,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Xd),Mi3)
        | aa(nat,$o,ord_less(nat,Ma),Xd) )
     => ( aa(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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_3434_succ__min,axiom,
    ! [Deg: nat,Xd: nat,Mi3: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less(nat,Xd),Mi3)
       => ( 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = aa(nat,option(nat),some(nat),Mi3) ) ) ) ).

% succ_min
tff(fact_3435_pred__max,axiom,
    ! [Deg: nat,Ma: nat,Xd: nat,Mi3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less(nat,Ma),Xd)
       => ( 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = aa(nat,option(nat),some(nat),Ma) ) ) ) ).

% pred_max
tff(fact_3436_lesseq__shift,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Xd),Y)
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),Xd),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_3437_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_3438_and__nat__numerals_I3_J,axiom,
    ! [Xd: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xd))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_3439_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_3440_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_3441_and__nat__numerals_I4_J,axiom,
    ! [Xd: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xd))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_3442_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_3443_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_3444_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_3445_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_3446_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_3447_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) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_3448_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))) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_3449_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_3450_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_3451_word__log__esimps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_log_esimps(1)
tff(fact_3452_word__log__esimps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),zero_zero(word(A))),Xd) = zero_zero(word(A)) ) ).

% word_log_esimps(7)
tff(fact_3453_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_3454_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_3455_vebt__pred_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va2: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $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_3456_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat,Y: word(A)] :
          ( ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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)),Xd),Y))),Nb) ) ) ).

% word_unat_and_lt
tff(fact_3457_VEBT__internal_Oreplicatei_Ocases,axiom,
    ! [A: $tType,Xd: product_prod(nat,heap_Time_Heap(A))] :
      ( ! [X: heap_Time_Heap(A)] : Xd != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),product_Pair(nat,heap_Time_Heap(A),zero_zero(nat)),X)
     => ~ ! [N: nat,X: heap_Time_Heap(A)] : Xd != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),product_Pair(nat,heap_Time_Heap(A),aa(nat,nat,suc,N)),X) ) ).

% VEBT_internal.replicatei.cases
tff(fact_3458_word__and__max__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),A2: word(A)] :
          ( ( Xd = 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),Xd) = A2 ) ) ) ).

% word_and_max_word
tff(fact_3459_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),D2)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Deg3: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_3460_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_3461_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_3462_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Va2: nat] : vEBT_vebt_succ(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va2) = none(nat) ).

% vebt_succ.simps(3)
tff(fact_3463_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list(vEBT_VEBT),Va2: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va2),Vb) = none(nat) ).

% vebt_pred.simps(4)
tff(fact_3464_and__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Nb) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% and_nat_def
tff(fact_3465_xor__num_Ocases,axiom,
    ! [Xd: product_prod(num,num)] :
      ( ( Xd != aa(num,product_prod(num,num),product_Pair(num,num,one2),one2) )
     => ( ! [N: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,one2),bit0(N))
       => ( ! [N: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N))
         => ( ! [M2: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,bit0(M2)),one2)
           => ( ! [M2: num,N: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,bit0(M2)),bit0(N))
             => ( ! [M2: num,N: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,bit0(M2)),aa(num,num,bit1,N))
               => ( ! [M2: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M2)),one2)
                 => ( ! [M2: num,N: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M2)),bit0(N))
                   => ~ ! [M2: num,N: num] : Xd != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M2)),aa(num,num,bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_3466_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_3467_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_3468_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B3: $o,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),X)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : Xd != aa(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)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)),X) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3469_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_3470_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_3471_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_3472_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,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B3: $o] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),zero_zero(nat))
     => ( ! [A5: $o,B3: $o] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A5: $o,B3: $o,N: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))
         => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu2: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Uu2)
           => ( ! [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TreeList2,Summary2)),X)
             => ( ! [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),X)
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_3473_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))
     => ( ! [A5: $o,Uw2: $o] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A5: $o,B3: $o,Va: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT,Vb2: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Vb2)
           => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : Xd != aa(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)),V4),zero_zero(nat),Vd2,Ve2)),Vf2)
             => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : Xd != aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Vj2)
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_3474_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,B3: $o] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(B3))),zero_zero(nat))
     => ( ! [Uv2: $o,Uw2: $o,N: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va3: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Va3)
         => ( ! [V4: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : Xd != aa(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)),V4),zero_zero(nat),Vc,Vd2)),Ve2)
           => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : Xd != aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Vi2)
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_3475_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,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B3: $o,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),X)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),X)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),X)
         => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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,V4)),TreeList2,Summary2)),X)
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_3476_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,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B3: $o,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),X)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),X)
       => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X: nat] : Xd != aa(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)),V4),zero_zero(nat),Uy2,Uz2)),X)
         => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : Xd != aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),X)
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_3477_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xd: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xd != aa(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] : Xd != aa(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,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),X)
         => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : Xd != aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),X)
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT,X: nat] : Xd != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),X) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3478_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),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),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),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_3479_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),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_3480_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T,Xd))
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(T,Xd)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% minNull_delete_time_bound
tff(fact_3481_mintlistlength,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Nb)
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less(nat,Mi3),Ma)
          & ? [M2: nat] :
              ( ( aa(nat,option(nat),some(nat),M2) = vEBT_vebt_mint(Summary) )
              & aa(nat,$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),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_3482_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_3483_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),zero_zero(nat),TrLst,Smry),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),zero_zero(nat),TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_3484_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(T)
     => ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T),X_1) ) ).

% not_min_Null_member
tff(fact_3485_min__Null__member,axiom,
    ! [T: vEBT_VEBT,Xd: nat] :
      ( vEBT_VEBT_minNull(T)
     => ~ aa(nat,$o,vEBT_vebt_member(T),Xd) ) ).

% min_Null_member
tff(fact_3486_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(T)
     => ( vEBT_vebt_mint(T) = none(nat) ) ) ).

% minNullmin
tff(fact_3487_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none(nat) )
     => vEBT_VEBT_minNull(T) ) ).

% minminNull
tff(fact_3488_mint__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(T),Maxi) ) ) ).

% mint_member
tff(fact_3489_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Mini: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(T),Xd)
         => aa(nat,$o,ord_less_eq(nat,Mini),Xd) ) ) ) ).

% mint_corr_help
tff(fact_3490_mint__sound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T),Xd)
       => ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),Xd) ) ) ) ).

% mint_sound
tff(fact_3491_mint__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),Xd) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T),Xd) ) ) ).

% mint_corr
tff(fact_3492_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T,Xd))
       => aa(nat,$o,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(T,Xd)),one_one(nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_3493_misiz,axiom,
    ! [T: vEBT_VEBT,Nb: nat,M: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(T) )
       => 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_3494_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_3495_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_3496_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xd)
     => ( ( Xd != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xd != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3497_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xd)
      <=> (Y) )
     => ( ( ( Xd = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xd = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xd = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3498_insertsimp,axiom,
    ! [T: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_minNull(T)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(T,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_3499_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_3500_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = vEBT_Leaf($false,(B2)) ).

% vebt_delete.simps(1)
tff(fact_3501_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Uu) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ).

% vebt_delete.simps(4)
tff(fact_3502_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_3503_nested__mint,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Nb: nat,Va2: 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Nb)
     => ( ( Nb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
       => ( ~ aa(nat,$o,ord_less(nat,Ma),Mi3)
         => ( ( Ma != Mi3 )
           => aa(nat,$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(Summary))),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))),divide_divide(nat,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)))))),aa(nat,nat,suc,divide_divide(nat,Va2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_3504_vebt__mint_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xd) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ( Y != $ite(
                  (A5),
                  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] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3505_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_3506_greater__shift,axiom,
    ! [Y: nat,Xd: nat] :
      ( aa(nat,$o,ord_less(nat,Y),Xd)
    <=> vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Xd),aa(nat,option(nat),some(nat),Y)) ) ).

% greater_shift
tff(fact_3507_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_3508_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_3509_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_3510_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_3511_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_3512_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,Xd: 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,Xd),Xaa),Xba) = Y )
     => ( ( ( Xaa = none(A) )
         => ( Y != none(A) ) )
       => ( ( ? [V4: A] : Xaa = aa(A,option(A),some(A),V4)
           => ( ( Xba = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A5: A] :
                ( ( Xaa = aa(A,option(A),some(A),A5) )
               => ! [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),Xd,A5),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_3513_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,Xd: product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,$o)),Uv2: option(A)] : Xd != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A)),Uu2),aa(option(A),product_prod(option(A),option(A)),product_Pair(option(A),option(A),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,$o)),V4: A] : Xd != aa(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)),product_Pair(option(A),option(A),aa(A,option(A),some(A),V4)),none(A)))
       => ~ ! [F4: fun(A,fun(A,$o)),X: A,Y2: A] : Xd != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A)),F4),aa(option(A),product_prod(option(A),option(A)),product_Pair(option(A),option(A),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_3514_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,Xd: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv2: option(A)] : Xd != aa(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)),product_Pair(option(A),option(A),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,A)),V4: A] : Xd != aa(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)),product_Pair(option(A),option(A),aa(A,option(A),some(A),V4)),none(A)))
       => ~ ! [F4: fun(A,fun(A,A)),A5: A,B3: A] : Xd != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A)),F4),aa(option(A),product_prod(option(A),option(A)),product_Pair(option(A),option(A),aa(A,option(A),some(A),A5)),aa(A,option(A),some(A),B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_3515_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_3516_summaxma,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi3 != Ma )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summary)) = vEBT_VEBT_high(Ma,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_3517_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_3518_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_3519_mul__shift,axiom,
    ! [Xd: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xd),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),Xd)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% mul_shift
tff(fact_3520_maxbmo,axiom,
    ! [T: vEBT_VEBT,Xd: nat] :
      ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),Xd) )
     => aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xd) ) ).

% maxbmo
tff(fact_3521_add__shift,axiom,
    ! [Xd: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xd),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),Xd)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% add_shift
tff(fact_3522_maxt__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(T),Maxi) ) ) ).

% maxt_member
tff(fact_3523_list__update__overwrite,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xd: A,Y: A] : list_update(A,list_update(A,Xs,I,Xd),I,Y) = list_update(A,Xs,I,Y) ).

% list_update_overwrite
tff(fact_3524_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Maxi: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(T),Xd)
         => aa(nat,$o,ord_less_eq(nat,Xd),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_3525_maxt__sound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T),Xd)
       => ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),Xd) ) ) ) ).

% maxt_sound
tff(fact_3526_maxt__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),Xd) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T),Xd) ) ) ).

% maxt_corr
tff(fact_3527_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xd: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I,Xd)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_3528_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_3529_nth__list__update__neq,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A),Xd: A] :
      ( ( I != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xd)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_3530_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xd: A] :
      ( aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( list_update(A,Xs,I,Xd) = Xs ) ) ).

% list_update_beyond
tff(fact_3531_nth__update__invalid,axiom,
    ! [A: $tType,I: nat,L: list(A),J: nat,Xd: A] :
      ( ~ aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,list_update(A,L,J,Xd)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_update_invalid
tff(fact_3532_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xd: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xd)),I) = Xd ) ) ).

% nth_list_update_eq
tff(fact_3533_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_3534_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mi3),Xd)
        & aa(nat,$o,ord_less_eq(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ~ vEBT_VEBT_minNull(Newnode)
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                   => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),
                                $ite(Xd = 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))),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)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_3535_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xd = Mi3 )
        & aa(nat,$o,ord_less(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,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(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( ~ vEBT_VEBT_minNull(Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_3536_list__update__swap,axiom,
    ! [A: $tType,I: nat,I5: nat,Xs: list(A),Xd: A,X6: A] :
      ( ( I != I5 )
     => ( list_update(A,list_update(A,Xs,I,Xd),I5,X6) = list_update(A,list_update(A,Xs,I5,X6),I,Xd) ) ) ).

% list_update_swap
tff(fact_3537_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),Xd: A,I: nat] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xs)),A3)
     => ( member(A,Xd,A3)
       => aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xd))),A3) ) ) ).

% set_update_subsetI
tff(fact_3538_in__set__upd__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xd: A,Y: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xd,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xd = Y )
          | ( member(A,Xd,aa(list(A),set(A),set2(A),L))
            & ! [Y5: A] : member(A,Xd,aa(list(A),set(A),set2(A),list_update(A,L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3539_set__update__memI,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Xd: A] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xd,aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xd))) ) ).

% set_update_memI
tff(fact_3540_in__set__upd__cases,axiom,
    ! [A: $tType,Xd: A,L: list(A),I: nat,Y: A] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
     => ( ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
         => ( Xd != Y ) )
       => member(A,Xd,aa(list(A),set(A),set2(A),L)) ) ) ).

% in_set_upd_cases
tff(fact_3541_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I: nat,L: list(A),Xd: A,Y: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xd,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xd = Y )
          | ! [Y5: A] : member(A,Xd,aa(list(A),set(A),set2(A),list_update(A,L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3542_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xd: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xd) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xd ) ) ) ).

% list_update_same_conv
tff(fact_3543_nth__list__update_H,axiom,
    ! [A: $tType,L: list(A),I: nat,Xd: A,J: nat] :
      aa(nat,A,nth(A,list_update(A,L,I,Xd)),J) = $ite(
        ( ( I = J )
        & aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L)) ),
        Xd,
        aa(nat,A,nth(A,L),J) ) ).

% nth_list_update'
tff(fact_3544_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xd: A,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xd)),J) = $ite(I = J,Xd,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_3545_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_3546_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_3547_vebt__maxt_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xd) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ( Y != $ite(
                  (B3),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3548_insert__simp__norm,axiom,
    ! [Xd: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi3: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,ord_less(nat,Mi3),Xd)
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xd != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),aa(nat,nat,ord_max(nat,Xd),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_3549_insert__simp__excp,axiom,
    ! [Mi3: nat,Deg: nat,TreeList: list(vEBT_VEBT),Xd: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Mi3,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,ord_less(nat,Xd),Mi3)
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xd != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xd),aa(nat,nat,ord_max(nat,Mi3),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi3,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi3,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi3,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi3,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Mi3,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_3550_less__shift,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xd),Y)
    <=> vEBT_VEBT_less(aa(nat,option(nat),some(nat),Xd),aa(nat,option(nat),some(nat),Y)) ) ).

% less_shift
tff(fact_3551_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_3552_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_3553_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_3554_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_3555_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,ord_max(nat,A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_3556_max__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),Nb) = Nb ).

% max_0L
tff(fact_3557_max__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,ord_max(nat,Nb),zero_zero(nat)) = Nb ).

% max_0R
tff(fact_3558_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_3559_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : aa(A,A,ord_max(A,zero_zero(A)),aa(num,A,numeral_numeral(A),Xd)) = aa(num,A,numeral_numeral(A),Xd) ) ).

% max_0_1(3)
tff(fact_3560_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),Xd)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xd) ) ).

% max_0_1(4)
tff(fact_3561_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_3562_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_3563_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_3564_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : aa(A,A,ord_max(A,one_one(A)),aa(num,A,numeral_numeral(A),Xd)) = aa(num,A,numeral_numeral(A),Xd) ) ).

% max_0_1(5)
tff(fact_3565_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),Xd)),one_one(A)) = aa(num,A,numeral_numeral(A),Xd) ) ).

% max_0_1(6)
tff(fact_3566_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_3567_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_3568_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_3569_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_3570_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_3571_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,ord_max(nat,Xd),Y)) = aa(A,A,ord_max(A,aa(nat,A,semiring_1_of_nat(A),Xd)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_3572_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,ord_max(A,Xd),Y)),Z) = aa(A,A,ord_max(A,aa(A,A,minus_minus(A,Xd),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_3573_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,ord_max(A,Xd),Y)),Z) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_3574_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),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),Xd),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Z)) ) ).

% max_add_distrib_right
tff(fact_3575_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_3576_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_3577_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_3578_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_3579_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_3580_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,ord_max(A,Xd),Y)),Z)
        <=> ( aa(A,$o,ord_less(A,Xd),Z)
            & aa(A,$o,ord_less(A,Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_3581_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_3582_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_3583_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi3: nat,Xd: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less_eq(nat,Mi3),Xd)
       => ( 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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xd,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)),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))),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,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))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_3584_succ__less__length__list,axiom,
    ! [Deg: nat,Mi3: nat,Xd: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less_eq(nat,Mi3),Xd)
       => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xd,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,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))),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,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))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_3585_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect(nat,vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_3586_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,ord_max(extended_enat,Q2),zero_zero(extended_enat)) = Q2 ).

% max_enat_simps(2)
tff(fact_3587_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,ord_max(extended_enat,zero_zero(extended_enat)),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_3588_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_aa(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_3589_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(set(complex),$o,finite_finite2(complex),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_ab(nat,fun(complex,fun(complex,$o)),Nb),C2))) ) ).

% finite_nth_roots
tff(fact_3590_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_ac(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_3591_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_ad(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_3592_del__x__not__mi,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mi3),Xd)
        & aa(nat,$o,ord_less_eq(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                 => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                   => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $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),product_Pair(nat,nat,Mi3),
                                  $ite(
                                    Xd = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi3,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))),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)))))) ),
                                    Ma ))),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),product_Pair(nat,nat,Mi3),
                                $ite(Xd = 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))),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)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_3593_del__x__not__mi__new__node__nil,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( aa(nat,$o,ord_less(nat,Mi3),Xd)
        & aa(nat,$o,ord_less_eq(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( vEBT_VEBT_minNull(Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),
                                  $ite(
                                    Xd = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none(nat),Mi3,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))),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)))))) ),
                                    Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_3594_del__x__not__mia,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mi3),Xd)
        & aa(nat,$o,ord_less_eq(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
               => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(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),product_Pair(nat,nat,Mi3),
                                  $ite(
                                    Xd = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi3,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))),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)))))) ),
                                    Ma ))),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),product_Pair(nat,nat,Mi3),
                                $ite(Xd = 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))),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)))),Ma))),Deg,newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_3595_del__x__mia,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & aa(nat,$o,ord_less(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $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(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(xn,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)),TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,divide_divide(nat,Deg,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),product_Pair(nat,nat,xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))))) ),
                                    Ma ))),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),product_Pair(nat,nat,xn),
                                $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))),Ma))),Deg,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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_3596_del__x__mi__lets__in__minNull,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & aa(nat,$o,ord_less(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,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(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_VEBT_minNull(Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary,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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))))) ),
                                      Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_3597_del__x__mi__lets__in,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xd = Mi3 )
        & aa(nat,$o,ord_less(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,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(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $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),product_Pair(nat,nat,Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))))) ),
                                      Ma ))),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),product_Pair(nat,nat,Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_3598_del__x__mi,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( Xd = Mi3 )
        & aa(nat,$o,ord_less(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,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(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(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),product_Pair(nat,nat,Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))))) ),
                                      Ma ))),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),product_Pair(nat,nat,Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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)))),Ma))),Deg,newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_3599_del__in__range,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less_eq(nat,Mi3),Xd)
        & aa(nat,$o,ord_less_eq(nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa(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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $let(
                xn: nat,
                xn:= 
                  $ite(Xd = Mi3,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))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xd),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xd = Mi3,xn,Mi3),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(xn,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)),TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,divide_divide(nat,Deg,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),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))),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)))))) ),
                                      Ma ))),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),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))),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)))),Ma))),Deg,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),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_3600_pred__lesseq__max,axiom,
    ! [Deg: nat,Xd: nat,Ma: nat,Mi3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less_eq(nat,Xd),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xd,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)),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))),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,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,ord_less(nat,Mi3),Xd),aa(nat,option(nat),some(nat),Mi3),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))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_3601_pred__less__length__list,axiom,
    ! [Deg: nat,Xd: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi3: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less_eq(nat,Xd),Ma)
       => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Deg,TreeList,Summary),Xd) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xd,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xd,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,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))),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,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,ord_less(nat,Mi3),Xd),aa(nat,option(nat),some(nat),Mi3),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))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_3602_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa: A] :
          aa(A,A,ord_max(A,X4),Xa) = $ite(aa(A,$o,ord_less_eq(A,X4),Xa),Xa,X4) ) ).

% max_def_raw
tff(fact_3603_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_3604_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect(nat,vEBT_V8194947554948674370ptions(T)) ).

% set_vebt_def
tff(fact_3605_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_ae(A,fun(A,$o),A2))),collect(A,aTP_Lamp_ae(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,dvd_dvd(A,A2),B2)
            & ~ aa(A,$o,dvd_dvd(A,B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3606_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_af(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_3607_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ag(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_3608_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_ah(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_3609_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),collect(A,aTP_Lamp_ae(A,fun(A,$o),A2))),collect(A,aTP_Lamp_ae(A,fun(A,$o),B2)))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_3610_less__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B4)
    <=> aa(fun(A,$o),$o,ord_less(fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B4)) ) ).

% less_set_def
tff(fact_3611_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),collect(int,aTP_Lamp_ai(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_3612_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aj(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_3613_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,ord_less(nat,X4),Xa)
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_3614_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,ord_less_eq(nat,X4),Xa)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_3615_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_3616_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_3617_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_3618_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X4),Xa) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_3619_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => aa(set(nat),$o,finite_finite2(nat),collect(nat,aTP_Lamp_ak(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_3620_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa) = nat2(aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_3621_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,minus_minus(nat,X4),Xa) = nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_3622_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : divide_divide(nat,X4,Xa) = nat2(divide_divide(int,aa(nat,int,semiring_1_of_nat(int),X4),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_3623_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : modulo_modulo(nat,X4,Xa) = nat2(modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X4),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_3624_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Nb)
         => aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_al(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_3625_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_am(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_eq
tff(fact_3626_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z4)) = $ite(
        aa(int,$o,ord_less(int,Z4),zero_zero(int)),
        nat2(Z),
        $let(
          d: int,
          d:= aa(int,int,minus_minus(int,Z),Z4),
          $ite(aa(int,$o,ord_less(int,d),zero_zero(int)),zero_zero(nat),nat2(d)) ) ) ).

% diff_nat_eq_if
tff(fact_3627_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_an(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_le
tff(fact_3628_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),divide_divide(word(A),aa(word(A),word(A),ord_max(word(A),A2),B2),C2)) = 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_3629_set__decode__def,axiom,
    ! [Xd: nat] : nat_set_decode(Xd) = collect(nat,aTP_Lamp_ao(nat,fun(nat,$o),Xd)) ).

% set_decode_def
tff(fact_3630_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_ap(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_3631_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        aa(nat,$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:= 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:= 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.simps(3)
tff(fact_3632_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,
    ! [Va2: nat] :
      vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        aa(nat,$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:= 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:= 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.simps(3)
tff(fact_3633_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,
    ! [Va2: nat] :
      vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        aa(nat,$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:= 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:= 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.simps(3)
tff(fact_3634_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,Xd: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList,S),Xd)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xd,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xd,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_3635_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,
    ! [Xd: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xd) = Y )
     => ( ( ( Xd = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [Va: nat] :
                ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_3636_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,
    ! [Xd: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xd) = Y )
     => ( ( ( Xd = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va: nat] :
                ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_3637_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,Xd: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList,Vd),Xd)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xd,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xd,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3638_vebt__buildup_Oelims,axiom,
    ! [Xd: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xd) = Y )
     => ( ( ( Xd = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3639_vebt__member_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xd)
    <=> $ite(
          Xd = Mi3,
          $true,
          $ite(
            Xd = Ma,
            $true,
            $ite(
              aa(nat,$o,ord_less(nat,Xd),Mi3),
              $false,
              $ite(
                aa(nat,$o,ord_less(nat,Ma),Xd),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xd,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,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xd,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_3640_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi3: nat,Ma: nat,V: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,V),TreeList,Vc2),Xd)
    <=> ( ( Xd = Mi3 )
        | ( Xd = Ma )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xd,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xd,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3641_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xd,Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xd != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xd = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3642_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xd,Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : Xd = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3643_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xd,Xaa)
      <=> (Y) )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xd = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xd = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3644_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),aa(A,A,ord_max(A,Xd),Y))
        <=> ( aa(A,$o,ord_less(A,Z),Xd)
            | aa(A,$o,ord_less(A,Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_3645_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_3646_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_3647_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_3648_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_3649_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xd,Xaa)
     => ( ! [Mi: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3650_vebt__member_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xd),Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa(nat,$o,ord_less(nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa(nat,$o,ord_less(nat,Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3651_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xd,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3652_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xd,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xd != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xd != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3653_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,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(
          Xd = Mi3,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xd = 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,Xd),Mi3),
                  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),Xd),
                      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(Xd,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(Xd,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.simps(5)
tff(fact_3654_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,ord_less(nat,Xd),Mi3),Mi3,Xd),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,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))
            & ~ ( ( Xd = Mi3 )
                | ( Xd = Ma ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  product_Pair(nat,nat,
                    $ite(aa(nat,$o,ord_less(nat,Xd),Mi3),Xd,Mi3)),
                  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,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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_3655_vebt__member_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xd),Xaa)
      <=> (Y) )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)
             => (Y) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
               => (Y) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3656_vebt__member_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xd),Xaa)
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xd != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)
           => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xd != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3657_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xd,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(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] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
         => ( ( ? [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( 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 = Ma2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Xaa),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,Ma2),Xaa),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_3658_vebt__insert_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xd,Xaa) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B3))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts2,S2) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ) )
           => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,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)),TreeList2))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma2 ) ) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),
                                    product_Pair(nat,nat,
                                      $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Xaa,Mi)),
                                    aa(nat,nat,ord_max(nat,xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3659_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $ite(
        aa(nat,$o,ord_less(nat,Ma),Xd),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xd,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(Xd,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'.simps(7)
tff(fact_3660_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $ite(
        aa(nat,$o,ord_less(nat,Xd),Mi3),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xd,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(Xd,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'.simps(6)
tff(fact_3661_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,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $ite(
        ( aa(nat,$o,ord_less(nat,Xd),Mi3)
        | aa(nat,$o,ord_less(nat,Ma),Xd) ),
        one_one(nat),
        $ite(
          ( ( Xd = Mi3 )
          & ( Xd = Ma ) ),
          one_one(nat),
          $let(
            xn: nat,
            xn:= 
              $ite(Xd = Mi3,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))),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)))))),Xd),
            $let(
              l: nat,
              l:= vEBT_VEBT_low(xn,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,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'.simps(7)
tff(fact_3662_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $ite(
        aa(nat,$o,ord_less(nat,Ma),Xd),
        aa(nat,option(nat),some(nat),Ma),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xd,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(Xd,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))),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,Mi3),Xd),aa(nat,option(nat),some(nat),Mi3),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))),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.simps(7)
tff(fact_3663_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $ite(
        aa(nat,$o,ord_less(nat,Xd),Mi3),
        aa(nat,option(nat),some(nat),Mi3),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xd,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(Xd,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))),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))),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.simps(6)
tff(fact_3664_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $ite(
        ( aa(nat,$o,ord_less(nat,Xd),Mi3)
        | aa(nat,$o,ord_less(nat,Ma),Xd) ),
        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
        $ite(
          ( ( Xd = Mi3 )
          & ( Xd = Ma ) ),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
          $let(
            xn: nat,
            xn:= 
              $ite(Xd = Mi3,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))),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)))))),Xd),
            $let(
              minn: nat,
              minn:= 
                $ite(Xd = Mi3,xn,Mi3),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,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,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),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))),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),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))),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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_3665_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xd,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,Uw2: $o] : Xd = vEBT_Leaf((A5),(Uw2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
             => ( ? [Va: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,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(Xaa,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)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
                                    one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_3666_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xd,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xd = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xd = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,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(Xaa,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)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_3667_vebt__delete_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xd,Xaa) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != vEBT_Leaf($false,(B3)) ) ) )
       => ( ! [A5: $o] :
              ( ? [B3: $o] : Xd = vEBT_Leaf((A5),(B3))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf((A5),$false) ) ) )
         => ( ! [A5: $o,B3: $o] :
                ( ( Xd = vEBT_Leaf((A5),(B3)) )
               => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
                 => ( Y != vEBT_Leaf((A5),(B3)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) ) )
             => ( ! [Mi: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                   => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TrLst2,Smry2) ) )
               => ( ! [Mi: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                     => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                              | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma2 ) ),
                                vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                $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(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi,xn,Mi),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,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)),TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                          $let(
                                            newlist: list(vEBT_VEBT),
                                            newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                            $ite(
                                              vEBT_VEBT_minNull(newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                    aa(nat,product_prod(nat,nat),product_Pair(nat,nat,minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $let(
                                                          maxs: option(nat),
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                                        Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                  aa(nat,product_prod(nat,nat),product_Pair(nat,nat,minn),
                                                    $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                                        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_3668_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xd,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
             => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                              | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                              one_one(nat),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma2 ) ),
                                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(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,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)),TreeList2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,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_3669_vebt__pred_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xd,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A5: $o] :
              ( ? [Uw2: $o] : Xd = vEBT_Leaf((A5),(Uw2))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A5: $o,B3: $o] :
                ( ( Xd = vEBT_Leaf((A5),(B3)) )
               => ( ? [Va: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
                 => ( Y != $ite(
                        (B3),
                        aa(nat,option(nat),some(nat),one_one(nat)),
                        $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2)
                 => ( Y != none(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              aa(nat,option(nat),some(nat),Ma2),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,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(Xaa,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)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary2,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))),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,TreeList2),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                    none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_3670_vebt__succ_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xd,Xaa) = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( Xd = 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] : Xd = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( 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,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(Xaa,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)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      $let(
                                        sc: option(nat),
                                        sc:= vEBT_vebt_succ(Summary2,h),
                                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_3671_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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(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_3672_lowi__hT,axiom,
    ! [Xd: nat,Nb: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_lowi(Xd,Nb),aa(nat,fun(nat,assn),aTP_Lamp_aq(nat,fun(nat,fun(nat,assn)),Xd),Nb),one_one(nat)) ).

% lowi_hT
tff(fact_3673_highi__hT,axiom,
    ! [Xd: nat,Nb: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_highi(Xd,Nb),aa(nat,fun(nat,assn),aTP_Lamp_ar(nat,fun(nat,fun(nat,assn)),Xd),Nb),one_one(nat)) ).

% highi_hT
tff(fact_3674_div__half__nat,axiom,
    ! [Y: nat,Xd: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,product_prod(nat,nat),product_Pair(nat,nat,divide_divide(nat,Xd,Y)),modulo_modulo(nat,Xd,Y)) = $let(
            q: nat,
            q:= aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,divide_divide(nat,Xd,aa(num,nat,numeral_numeral(nat),bit0(one2))),Y)),
            $let(
              r: nat,
              r:= aa(nat,nat,minus_minus(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),q),Y)),
              $ite(aa(nat,$o,ord_less_eq(nat,Y),r),aa(nat,product_prod(nat,nat),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),product_Pair(nat,nat,q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_3675_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xd,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(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),S2: vEBT_VEBT] : Xd = vEBT_Node(Info2,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xd = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( 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,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)),TreeList2))
                                & ~ ( ( Xaa = Mi )
                                    | ( Xaa = Ma2 ) ) ),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
                                one_one(nat) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_3676_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xd,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma2,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,ord_less(nat,Mi),Xaa)
                                    & aa(nat,$o,ord_less(nat,Xaa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                    zero_zero(nat) ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_3677_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,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = 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,Xd),Mi3),Mi3,Xd),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,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))
              & ~ ( ( Xd = Mi3 )
                  | ( Xd = 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,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.simps(5)
tff(fact_3678_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xd,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xd = vEBT_Node(Info2,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xd = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,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)),TreeList2))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma2 ) ) ),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_3679_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,Xd: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A2),(B2)),Xd) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_3680_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : aa(nat,$o,ord_less_eq(nat,vEBT_T_m_i_n_N_u_l_l(T)),one_one(nat)) ).

% minNull_bound
tff(fact_3681_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_3682_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_3683_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_3684_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,Xd: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A2),(B2)),Xd) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_3685_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),S: vEBT_VEBT,Xd: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts,S),Xd) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_3686_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),Va2: 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),Va2,Vb,Vc2)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_3687_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_3688_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,Xd: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xd) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_3689_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),S: vEBT_VEBT,Xd: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xd) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_3690_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary),Xd) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_3691_insersimp_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T),X_1)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(T,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_3692_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_minNull(T)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(T,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_3693_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,Xd: 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),Xd) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_3694_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(T,Xd)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% insert'_bound_height
tff(fact_3695_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,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xd) = Y )
     => ( ( ( Xd = vEBT_Leaf($false,$false) )
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uv2: $o] : Xd = vEBT_Leaf($true,(Uv2))
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu2: $o] : Xd = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ( Y != one_one(nat) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_3696_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,Xd: 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),Xd) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_3697_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_m_e_m_b_e_r2(T,Xd)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% member_bound_height'
tff(fact_3698_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,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,ord_less(nat,Xd),Mi3),Mi3,Xd),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,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))
            & ~ ( ( Xd = Mi3 )
                | ( Xd = 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,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'.simps(5)
tff(fact_3699_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,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xd = Mi3,
          zero_zero(nat),
          $ite(
            Xd = Ma,
            zero_zero(nat),
            $ite(
              aa(nat,$o,ord_less(nat,Xd),Mi3),
              zero_zero(nat),
              $ite(
                aa(nat,$o,ord_less(nat,Ma),Xd),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,ord_less(nat,Mi3),Xd)
                  & aa(nat,$o,ord_less(nat,Xd),Ma) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xd,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(Xd,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'.simps(5)
tff(fact_3700_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aTP_Lamp_as(nat,fun(nat,$o),K))) ).

% finite_Collect_less_nat
tff(fact_3701_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xd,Xaa) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
             => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( 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,Ma2),Xaa) ),
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma2 ) ),
                                    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                    aa(nat,nat,
                                      aa(nat,fun(nat,nat),plus_plus(nat),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                              $ite(Xaa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                          one_one(nat))),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(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(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(xn,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)),TreeList2)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                  $let(
                                                    newlist: list(vEBT_VEBT),
                                                    newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                      $ite(
                                                        vEBT_VEBT_minNull(newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary2,h),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                  $let(
                                                                    maxs: option(nat),
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                      $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                                                one_one(nat) )) )),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                          $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                              one_one(nat) ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_3702_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,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = 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,Xd),Mi3)
          | aa(nat,$o,ord_less(nat,Ma),Xd) ),
          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(
              ( ( Xd = Mi3 )
              & ( Xd = 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(Xd = Mi3,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(Xd = Mi3,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))),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)))))),Xd),
                  $let(
                    l: nat,
                    l:= vEBT_VEBT_low(xn,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,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.simps(7)
tff(fact_3703_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xd,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xd = 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] : Xd = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( 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,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                    $ite(
                                      aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                        $let(
                                          maxlow: option(nat),
                                          maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                            $ite(
                                              ( ( maxlow != none(nat) )
                                              & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                sc: option(nat),
                                                sc:= vEBT_vebt_succ(Summary2,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                                                  $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_3704_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_3705_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_3706_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_3707_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : aa(nat,$o,ord_less_eq(nat,vEBT_T_m_a_x_t(T)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_3708_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi3: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_3709_mint__bound,axiom,
    ! [T: vEBT_VEBT] : aa(nat,$o,ord_less_eq(nat,vEBT_T_m_i_n_t(T)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_3710_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_3711_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi3: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_3712_finite__psubset__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [A8: set(A)] :
            ( aa(set(A),$o,finite_finite2(A),A8)
           => ( ! [B8: set(A)] :
                  ( aa(set(A),$o,ord_less(set(A),B8),A8)
                 => aa(set(A),$o,P,B8) )
             => aa(set(A),$o,P,A8) ) )
       => aa(set(A),$o,P,A3) ) ) ).

% finite_psubset_induct
tff(fact_3713_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xd) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(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] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_3714_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xd) = Y )
     => ( ! [A5: $o] :
            ( ? [B3: $o] : Xd = vEBT_Leaf((A5),(B3))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A5),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] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_3715_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Ma),Xd),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xd,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(Xd,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.simps(7)
tff(fact_3716_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xd: 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),product_Pair(nat,nat,Mi3),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xd) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Xd),Mi3),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xd,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(Xd,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.simps(6)
tff(fact_3717_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xd,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,Uw2: $o] : Xd = vEBT_Leaf((A5),(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)) ) ) )
         => ( ! [A5: $o,B3: $o] :
                ( ( Xd = vEBT_Leaf((A5),(B3)) )
               => ( ? [Va: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
                 => ( 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),Va3: vEBT_VEBT] : Xd = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                            $ite(
                                              ( ( minlow != none(nat) )
                                              & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                pr: option(nat),
                                                pr:= vEBT_vebt_pred(Summary2,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                                                  $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                        one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_3718_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,B3: $o] :
                ( ( Xd = vEBT_Leaf((A5),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B3: $o] :
                  ( ( Xd = vEBT_Leaf((A5),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( 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,Ma2),Xaa) ),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                      $ite(
                                        ( ( Xaa = Mi )
                                        & ( Xaa = Ma2 ) ),
                                        aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,
                                              aa(nat,fun(nat,nat),plus_plus(nat),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                                  $ite(Xaa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                              one_one(nat))),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(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(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(xn,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)),TreeList2)),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                      $let(
                                                        newlist: list(vEBT_VEBT),
                                                        newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                          $ite(
                                                            vEBT_VEBT_minNull(newnode),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                      $let(
                                                                        maxs: option(nat),
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                          $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                                                    one_one(nat) )) )),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                                  one_one(nat) ) ) ) )) )) )) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_3719_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xd = 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)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( 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,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                        $ite(
                                          aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                            $let(
                                              maxlow: option(nat),
                                              maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                                $ite(
                                                  ( ( maxlow != none(nat) )
                                                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    sc: option(nat),
                                                    sc:= vEBT_vebt_succ(Summary2,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_3720_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((A5),(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)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B3: $o] :
                  ( ( Xd = vEBT_Leaf((A5),(B3)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( 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)))) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                    one_one(nat),
                                    $let(
                                      l: nat,
                                      l:= vEBT_VEBT_low(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                          $ite(
                                            aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              minlow: option(nat),
                                              minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                                $ite(
                                                  ( ( minlow != none(nat) )
                                                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    pr: option(nat),
                                                    pr:= vEBT_vebt_pred(Summary2,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                                                      $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                            one_one(nat) )) ) ) )) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_3721_pred__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_at(vEBT_VEBT,fun(nat,fun(nat,$o)),T),Xd)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_3722_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_3723_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_mint(T) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_3724_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_maxt(T) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_3725_succ__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_au(vEBT_VEBT,fun(nat,fun(nat,$o)),T),Xd)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_3726_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_3727_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H),bot_bot(set(nat))))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H),bot_bot(set(nat))))
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q),aa(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_3728_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(pure_assn((B2))),aa(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_3729_mod__h__bot__iff_I4_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Q2: array(A),Y: list(A),H: heap_ext(product_unit)] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(list(A),assn,snga_assn(A,Q2),Y)),aa(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_3730_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3731_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_3732_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_ext(product_unit),H4: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H),bot_bot(set(nat))))
    <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H4),bot_bot(set(nat)))) ) ).

% mod_h_bot_indep
tff(fact_3733_not__psubset__empty,axiom,
    ! [A: $tType,A3: set(A)] : ~ aa(set(A),$o,ord_less(set(A),A3),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_3734_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,S3)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S3)
                      & aa(A,$o,ord_less(A,Xa),X) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_3735_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X7: set(A)] :
          ( ( X7 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X7)
               => ? [Xa: A] :
                    ( member(A,Xa,X7)
                    & aa(A,$o,ord_less(A,X),Xa) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X7) ) ) ) ).

% infinite_growing
tff(fact_3736_mod__emp__simp,axiom,
    ! [H: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(one_one(assn)),aa(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_3737_vebt__succ_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(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)),V4),zero_zero(nat),Vc,Vd2)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( 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,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(Xaa,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)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary2,h),
                                            $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3738_vebt__pred_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((A5),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B3: $o] :
                  ( ( Xd = vEBT_Leaf((A5),(B3)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = $ite(
                              (B3),
                              aa(nat,option(nat),some(nat),one_one(nat)),
                              $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = none(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  aa(nat,option(nat),some(nat),Ma2),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,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(Xaa,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)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary2,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))),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,TreeList2),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                        none(nat) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_3739_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,B3: $o] :
                ( ( Xd = vEBT_Leaf((A5),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B3: $o] :
                  ( ( Xd = vEBT_Leaf((A5),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                                  | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                                  one_one(nat),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma2 ) ),
                                    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(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,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)),TreeList2)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                                            one_one(nat) ) ) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_3740_vebt__delete_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf($false,(B3)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,B3: $o] :
                ( ( Xd = vEBT_Leaf((A5),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf((A5),$false) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B3: $o] :
                  ( ( Xd = vEBT_Leaf((A5),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = vEBT_Leaf((A5),(B3)) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),TrLst2,Smry2)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                                  | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma2 ) ),
                                    vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                    $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(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi,xn,Mi),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,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)),TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                              $let(
                                                newlist: list(vEBT_VEBT),
                                                newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                $ite(
                                                  vEBT_VEBT_minNull(newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary2,h),
                                                    vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                        aa(nat,product_prod(nat,nat),product_Pair(nat,nat,minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $let(
                                                              maxs: option(nat),
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                                            Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                      aa(nat,product_prod(nat,nat),product_Pair(nat,nat,minn),
                                                        $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_3741_one__assn__raw_Ocases,axiom,
    ! [Xd: product_prod(heap_ext(product_unit),set(nat))] :
      ~ ! [H3: heap_ext(product_unit),As: set(nat)] : Xd != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H3),As) ).

% one_assn_raw.cases
tff(fact_3742_pure__assn__raw_Ocases,axiom,
    ! [A: $tType,B: $tType,Xd: product_prod($o,product_prod(A,set(B)))] :
      ~ ! [B3: $o,H3: A,As: set(B)] : Xd != aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),product_Pair($o,product_prod(A,set(B)),(B3)),aa(set(B),product_prod(A,set(B)),product_Pair(A,set(B),H3),As)) ).

% pure_assn_raw.cases
tff(fact_3743_snga__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xd: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R3: array(A),X: list(A),H3: heap_ext(product_unit),As: set(nat)] : Xd != 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)))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),R3),aa(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)),X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H3),As))) ) ).

% snga_assn_raw.cases
tff(fact_3744_sngr__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xd: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R3: ref(A),X: A,H3: heap_ext(product_unit),As: set(nat)] : Xd != 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)))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),R3),aa(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)),X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H3),As))) ) ).

% sngr_assn_raw.cases
tff(fact_3745_times__assn__raw_Ocases,axiom,
    ! [Xd: 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),H3: heap_ext(product_unit),As: set(nat)] : Xd != 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)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$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))),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)),product_Pair(heap_ext(product_unit),set(nat),H3),As))) ).

% times_assn_raw.cases
tff(fact_3746_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vc,Vd2)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,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(Xaa,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)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_3747_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(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)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( 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,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)),TreeList2))
                                    & ~ ( ( Xaa = Mi )
                                        | ( Xaa = Ma2 ) ) ),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_3748_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((A5),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B3: $o] :
                  ( ( Xd = vEBT_Leaf((A5),(B3)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,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(Xaa,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)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
                                        one_one(nat) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_3749_vebt__insert_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B3))) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(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,V4)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,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)),TreeList2))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma2 ) ) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                      aa(nat,product_prod(nat,nat),
                                        product_Pair(nat,nat,
                                          $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Xaa,Mi)),
                                        aa(nat,nat,ord_max(nat,xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3750_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(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,V4)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,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)),TreeList2))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_3751_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(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)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( 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 = Ma2,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,Xaa),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,Ma2),Xaa),
                                            one_one(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_3752_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,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xd,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Xaa),Mi),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,ord_less(nat,Mi),Xaa)
                                        & aa(nat,$o,ord_less(nat,Xaa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                        zero_zero(nat) ) ) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_3753_vebt__member_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xd),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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)),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xaa)) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3754_vebt__member_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xd),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3755_vebt__member_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xd),Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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)),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ( ~ (Y)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(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)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,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)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3756_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xd,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(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)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3757_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xd,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3758_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xd,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B3))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(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)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3759_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xd,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3760_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xd,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3761_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xd,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(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),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),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),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),Xaa)) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3762_monoseq__arctan__series,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_av(real,fun(nat,real),Xd)) ) ).

% monoseq_arctan_series
tff(fact_3763_delete__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xd)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(T)),insert(nat,Xd,bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_3764_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_aw(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_3765_foldr__zero,axiom,
    ! [Xs: list(nat),D: 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,ord_less_eq(nat,aa(list(nat),nat,size_size(list(nat)),Xs)),aa(nat,nat,minus_minus(nat,aa(nat,nat,foldr(nat,nat,plus_plus(nat),Xs),D)),D)) ) ).

% foldr_zero
tff(fact_3766_foldr__one,axiom,
    ! [D: nat,Ys: list(nat)] : aa(nat,$o,ord_less_eq(nat,D),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Ys),D)) ).

% foldr_one
tff(fact_3767_foldr__same__int,axiom,
    ! [Xs: list(nat),Y: nat] :
      ( ! [X: nat,Y2: nat] :
          ( member(nat,X,aa(list(nat),set(nat),set2(nat),Xs))
         => ( member(nat,Y2,aa(list(nat),set(nat),set2(nat),Xs))
           => ( X = Y2 ) ) )
     => ( ! [X: nat] :
            ( member(nat,X,aa(list(nat),set(nat),set2(nat),Xs))
           => ( X = Y ) )
       => ( aa(nat,nat,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_3768_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( vEBT_invar_vebt(T,Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xd)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_VEBT_set_vebt(T)),insert(nat,Xd,bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_3769_foldr__mono,axiom,
    ! [Xs: list(nat),Ys: list(nat),C2: nat,D: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys) )
     => ( ! [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,$o,ord_less_eq(nat,C2),D)
         => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Xs),C2)),aa(list(nat),nat,size_size(list(nat)),Ys))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Ys),D)) ) ) ) ).

% foldr_mono
tff(fact_3770_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : aa(nat,nat,foldr(A,nat,aTP_Lamp_ax(A,fun(nat,nat)),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_3771_set__replicate,axiom,
    ! [A: $tType,Nb: nat,Xd: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Nb,Xd)) = insert(A,Xd,bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_3772_set__encode__insert,axiom,
    ! [A3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( ~ member(nat,Nb,A3)
       => ( aa(set(nat),nat,nat_set_encode,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_3773_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A5: A,X: B] :
              ( member(B,X,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F2,X),A5) = aa(A,A,aa(B,fun(A,A),G,X),A5) ) )
         => ( aa(A,A,foldr(B,A,F2,L),A2) = aa(A,A,foldr(B,A,G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_3774_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A8)
                 => ( ! [X4: A] :
                        ( member(A,X4,A8)
                       => aa(A,$o,ord_less(A,B3),X4) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,insert(A,B3,A8)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3775_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A8)
                 => ( ! [X4: A] :
                        ( member(A,X4,A8)
                       => aa(A,$o,ord_less(A,X4),B3) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,insert(A,B3,A8)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3776_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = insert(nat,aa(nat,nat,suc,Nb),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_3777_atLeastAtMost__insertL,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( insert(nat,M,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) = set_or1337092689740270186AtMost(nat,M,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_3778_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Nb))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb)) = insert(nat,aa(nat,nat,suc,Nb),set_or1337092689740270186AtMost(nat,M,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3779_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( set_or1337092689740270186AtMost(nat,M,Nb) = insert(nat,M,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3780_remove__subset,axiom,
    ! [A: $tType,Xd: A,S3: set(A)] :
      ( member(A,Xd,S3)
     => aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),S3),insert(A,Xd,bot_bot(set(A))))),S3) ) ).

% remove_subset
tff(fact_3781_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xd: A] : aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xd))),insert(A,Xd,aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_3782_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A2: nat] : aa(nat,nat,foldr(A,nat,aTP_Lamp_ax(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_3783_finite__induct__select,axiom,
    ! [A: $tType,S3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T2: set(A)] :
              ( aa(set(A),$o,ord_less(set(A),T2),S3)
             => ( aa(set(A),$o,P,T2)
               => ? [X4: A] :
                    ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),S3),T2))
                    & aa(set(A),$o,P,insert(A,X4,T2)) ) ) )
         => aa(set(A),$o,P,S3) ) ) ) ).

% finite_induct_select
tff(fact_3784_set__replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xd: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Nb),Xd)) = insert(A,Xd,bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_3785_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xd: A,B4: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),insert(A,Xd,B4))
    <=> $ite(
          member(A,Xd,B4),
          aa(set(A),$o,ord_less(set(A),A3),B4),
          $ite(member(A,Xd,A3),aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A))))),B4),aa(set(A),$o,ord_less_eq(set(A),A3),B4)) ) ) ).

% psubset_insert_iff
tff(fact_3786_set__replicate__conv__if,axiom,
    ! [A: $tType,Nb: nat,Xd: A] :
      aa(list(A),set(A),set2(A),replicate(A,Nb,Xd)) = $ite(Nb = zero_zero(nat),bot_bot(set(A)),insert(A,Xd,bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_3787_insert__swap__set__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xd: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( insert(A,aa(nat,A,nth(A,L),I),aa(list(A),set(A),set2(A),list_update(A,L,I,Xd))) = insert(A,Xd,aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_3788_monoseq__realpow,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),Xd)) ) ) ).

% monoseq_realpow
tff(fact_3789_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)) = insert(nat,Nb,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_3790_foldr__same,axiom,
    ! [Xs: list(real),Y: real] :
      ( ! [X: real,Y2: real] :
          ( member(real,X,aa(list(real),set(real),set2(real),Xs))
         => ( member(real,Y2,aa(list(real),set(real),set2(real),Xs))
           => ( X = Y2 ) ) )
     => ( ! [X: real] :
            ( member(real,X,aa(list(real),set(real),set2(real),Xs))
           => ( X = Y ) )
       => ( aa(real,real,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_3791_diff__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),N2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_ay(fun(A,nat),fun(A,$o),M7)))
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,nat),fun(A,$o),aTP_Lamp_az(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M7),N2))) ) ).

% diff_preserves_multiset
tff(fact_3792_add__mset__in__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),A2: A] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_ay(fun(A,nat),fun(A,$o),M7)))
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_ba(fun(A,nat),fun(A,fun(A,$o)),M7),A2))) ) ).

% add_mset_in_multiset
tff(fact_3793_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),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_bb(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_3794_foldr0,axiom,
    ! [Xs: list(real),C2: real,D: real] : aa(real,real,foldr(real,real,plus_plus(real),Xs),aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,foldr(real,real,plus_plus(real),Xs),D)),C2) ).

% foldr0
tff(fact_3795_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M7) = zero_zero(nat) )
    <=> ( M7 = zero_zero(multiset(A)) ) ) ).

% size_eq_0_iff_empty
tff(fact_3796_size__empty,axiom,
    ! [A: $tType] : aa(multiset(A),nat,size_size(multiset(A)),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_3797_size__union,axiom,
    ! [A: $tType,M7: multiset(A),N2: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),M7)),aa(multiset(A),nat,size_size(multiset(A)),N2)) ).

% size_union
tff(fact_3798_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_3799_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_bc(fun(B,A),fun(A,fun(B,A)),F2),A2),A3),A2) = modulo_modulo(A,groups7121269368397514597t_prod(B,A,F2,A3),A2) ) ).

% mod_prod_eq
tff(fact_3800_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_bd(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3801_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_be(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3802_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_bf(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M),set_or1337092689740270186AtMost(nat,Nb,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3803_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_3804_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_3805_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_3806_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_bd(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3807_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_bg(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))) ) ).

% fact_prod
tff(fact_3808_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Nb: int] :
      ( aa(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)) = 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_3809_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)),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_3810_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : groups7121269368397514597t_prod(nat,A,F2,set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_bh(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3811_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),groups7121269368397514597t_prod(A,B,F2,A3))
          <=> ? [X5: A] :
                ( member(A,X5,A3)
                & aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F2,X5)) ) ) ) ) ).

% even_prod_iff
tff(fact_3812_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$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),P3))) = 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),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3813_fact__eq__fact__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(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_bg(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),M))) ) ) ).

% fact_eq_fact_times
tff(fact_3814_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_bi(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_3815_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_bj(A,fun(nat,fun(nat,A)),A2),Nb),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_3816_fact__div__fact,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Nb),M)
     => ( divide_divide(nat,semiring_char_0_fact(nat,M),semiring_char_0_fact(nat,Nb)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_bg(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_3817_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_bk(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.in_pairs
tff(fact_3818_nonempty__has__size,axiom,
    ! [A: $tType,S3: multiset(A)] :
      ( ( S3 != zero_zero(multiset(A)) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S3)) ) ).

% nonempty_has_size
tff(fact_3819_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_bj(A,fun(nat,fun(nat,A)),A2),Nb),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3820_filter__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_ay(fun(A,nat),fun(A,$o),M7)))
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bl(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M7),P))) ) ).

% filter_preserves_multiset
tff(fact_3821_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( member(int,K,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))
        & member(int,L,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% and_int.simps
tff(fact_3822_and__int_Oelims,axiom,
    ! [Xd: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xd),Xaa) = Y )
     => ( Y = $ite(
            ( member(int,Xd,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))
            & member(int,Xaa,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xd)
                & ~ aa(int,$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))),Xd)
                  & ~ 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),divide_divide(int,Xd,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xaa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_3823_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xd: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xd,A3)
           => ( groups7121269368397514597t_prod(A,B,G,insert(A,Xd,A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xd)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod.insert
tff(fact_3824_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bm(A,fun(fun(A,B),fun(A,B)),A2),B2),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_3825_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bn(A,fun(fun(A,B),fun(A,B)),A2),B2),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_3826_dvd__prodI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),A2: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => aa(B,$o,dvd_dvd(B,aa(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ) ).

% dvd_prodI
tff(fact_3827_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,Xd: multiset(A),Y: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xd),Y) = zero_zero(multiset(A)) )
    <=> ( ( Xd = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_3828_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType,Xd: 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)),Xd),Y) )
    <=> ( ( Xd = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_3829_empty__eq__union,axiom,
    ! [A: $tType,M7: multiset(A),N2: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N2) )
    <=> ( ( M7 = zero_zero(multiset(A)) )
        & ( N2 = zero_zero(multiset(A)) ) ) ) ).

% empty_eq_union
tff(fact_3830_union__eq__empty,axiom,
    ! [A: $tType,M7: multiset(A),N2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N2) = zero_zero(multiset(A)) )
    <=> ( ( M7 = zero_zero(multiset(A)) )
        & ( N2 = zero_zero(multiset(A)) ) ) ) ).

% union_eq_empty
tff(fact_3831_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_bo(B,A),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3832_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( groups7121269368397514597t_prod(A,B,F2,A3) = zero_zero(B) )
          <=> ? [X5: A] :
                ( member(A,X5,A3)
                & ( aa(A,B,F2,X5) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_3833_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_3834_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,A3) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_3835_dvd__prod__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),A2: A,B2: B,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => ( ( B2 = aa(A,B,F2,A2) )
             => aa(B,$o,dvd_dvd(B,B2),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_3836_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( groups7121269368397514597t_prod(A,nat,F2,A3) = one_one(nat) )
      <=> ! [X5: A] :
            ( member(A,X5,A3)
           => ( aa(A,nat,F2,X5) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3837_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),groups7121269368397514597t_prod(A,nat,F2,A3))
      <=> ! [X5: A] :
            ( member(A,X5,A3)
           => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,F2,X5)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3838_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_3839_diff__empty,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),zero_zero(multiset(A))) = M7 )
      & ( aa(multiset(A),multiset(A),minus_minus(multiset(A),zero_zero(multiset(A))),M7) = zero_zero(multiset(A)) ) ) ).

% diff_empty
tff(fact_3840_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: multiset(A),C6: multiset(A),B4: multiset(A),D3: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),A3),C6)
         => ( aa(multiset(A),$o,ord_less(multiset(A),B4),D3)
           => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C6),D3)) ) ) ) ).

% union_less_mono
tff(fact_3841_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D3: multiset(A),C6: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B4),D3)
         => 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)),C6),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C6),D3)) ) ) ).

% union_le_mono2
tff(fact_3842_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D3: multiset(A),C6: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B4),D3)
         => 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)),B4),C6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D3),C6)) ) ) ).

% union_le_mono1
tff(fact_3843_empty__neutral_I2_J,axiom,
    ! [A: $tType,Xd: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xd),zero_zero(multiset(A))) = Xd ).

% empty_neutral(2)
tff(fact_3844_empty__neutral_I1_J,axiom,
    ! [A: $tType,Xd: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),zero_zero(multiset(A))),Xd) = Xd ).

% empty_neutral(1)
tff(fact_3845_union__diff__assoc,axiom,
    ! [A: $tType,C6: multiset(A),B4: multiset(A),A3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C6),B4) = 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),B4)),C6) = 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),B4),C6)) ) ) ).

% union_diff_assoc
tff(fact_3846_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_bp(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_3847_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => ( aa(A,B,G,X) = one_one(B) ) )
         => ( groups7121269368397514597t_prod(A,B,G,A3) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_3848_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) )
         => ~ ! [A5: B] :
                ( member(B,A5,A3)
               => ( aa(B,A,G,A5) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3849_prod__dvd__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [A5: A] :
              ( member(A,A5,A3)
             => aa(B,$o,dvd_dvd(B,aa(A,B,F2,A5)),aa(A,B,G,A5)) )
         => aa(B,$o,dvd_dvd(B,groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ).

% prod_dvd_prod
tff(fact_3850_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_bq(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_3851_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_br(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A3) = divide_divide(A,groups7121269368397514597t_prod(B,A,F2,A3),groups7121269368397514597t_prod(B,A,G,A3)) ) ).

% prod_dividef
tff(fact_3852_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: set(B),Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),groups7121269368397514597t_prod(B,A,F2,A3)),Nb) = groups7121269368397514597t_prod(B,A,aa(nat,fun(B,A),aTP_Lamp_bs(fun(B,A),fun(nat,fun(B,A)),F2),Nb),A3) ) ).

% prod_power_distrib
tff(fact_3853_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X)) )
         => aa(B,$o,ord_less_eq(B,zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ).

% prod_nonneg
tff(fact_3854_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A3)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3))
                & aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
         => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ).

% prod_mono
tff(fact_3855_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,X)) )
         => aa(B,$o,ord_less(B,zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ).

% prod_pos
tff(fact_3856_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,F2,X)) )
         => aa(B,$o,ord_less_eq(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ).

% prod_ge_1
tff(fact_3857_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ? [X4: A] :
                ( member(A,X4,A3)
                & ( aa(A,B,F2,X4) = zero_zero(B) ) )
           => ( groups7121269368397514597t_prod(A,B,F2,A3) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_3858_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),Xd: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bt(set(A),fun(fun(A,B),fun(A,$o)),I6),Xd)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bt(set(A),fun(fun(A,B),fun(A,$o)),I6),Y)))
           => aa(set(A),$o,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_bu(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I6),Xd),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_3859_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),Xd: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bv(set(A),fun(fun(A,B),fun(A,$o)),I6),Xd)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bv(set(A),fun(fun(A,B),fun(A,$o)),I6),Y)))
           => aa(set(A),$o,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_bw(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I6),Xd),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3860_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = groups7121269368397514597t_prod(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_by(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P),A3) ) ) ) ).

% prod.inter_filter
tff(fact_3861_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X))
                & aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),one_one(B)) ) )
         => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A3)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3862_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X22),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S3)
             => ( ! [X: B] :
                    ( member(B,X,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X)),aa(B,A,G,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R,groups7121269368397514597t_prod(B,A,H,S3)),groups7121269368397514597t_prod(B,A,G,S3)) ) ) ) ) ) ).

% prod.related
tff(fact_3863_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xd: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,insert(A,Xd,A3)) = $ite(member(A,Xd,A3),groups7121269368397514597t_prod(A,B,G,A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xd)),groups7121269368397514597t_prod(A,B,G,A3))) ) ) ) ).

% prod.insert_if
tff(fact_3864_prod__dvd__prod__subset2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
           => ( ! [A5: A] :
                  ( member(A,A5,A3)
                 => aa(B,$o,dvd_dvd(B,aa(A,B,F2,A5)),aa(A,B,G,A5)) )
             => aa(B,$o,dvd_dvd(B,groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,G,B4)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_3865_prod__dvd__prod__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
           => aa(B,$o,dvd_dvd(B,groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,B4)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_3866_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T4: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( ! [A5: A] :
                  ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S3),S4))
                 => ( aa(B,A,I,aa(A,B,J,A5)) = A5 ) )
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S3),S4))
                   => member(B,aa(A,B,J,A5),aa(set(B),set(B),minus_minus(set(B),T4),T3)) )
               => ( ! [B3: B] :
                      ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T4),T3))
                     => ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T4),T3))
                       => member(A,aa(B,A,I,B3),aa(set(A),set(A),minus_minus(set(A),S3),S4)) )
                   => ( ! [A5: A] :
                          ( member(A,A5,S4)
                         => ( aa(A,C,G,A5) = one_one(C) ) )
                     => ( ! [B3: B] :
                            ( member(B,B3,T3)
                           => ( aa(B,C,H,B3) = one_one(C) ) )
                       => ( ! [A5: A] :
                              ( member(A,A5,S3)
                             => ( aa(B,C,H,aa(A,B,J,A5)) = aa(A,C,G,A5) ) )
                         => ( groups7121269368397514597t_prod(A,C,G,S3) = groups7121269368397514597t_prod(B,C,H,T4) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3867_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),collect(A,aTP_Lamp_bz(fun(A,B),fun(A,$o),G)))) = groups7121269368397514597t_prod(A,B,G,A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3868_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( member(A,I,I6)
           => ( aa(B,$o,ord_less(B,one_one(B)),aa(A,B,F2,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I6)
                   => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,I6)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3869_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ( I6 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I6)
                 => aa(B,$o,ord_less(B,one_one(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,I6)) ) ) ) ) ).

% less_1_prod
tff(fact_3870_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C6: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C6)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),C6)
           => ( aa(set(A),$o,ord_less_eq(set(A),B4),C6)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C6),A3))
                   => ( aa(A,B,G,A5) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C6),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,A3) = groups7121269368397514597t_prod(A,B,H,B4) )
                  <=> ( groups7121269368397514597t_prod(A,B,G,C6) = groups7121269368397514597t_prod(A,B,H,C6) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3871_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C6: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C6)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),C6)
           => ( aa(set(A),$o,ord_less_eq(set(A),B4),C6)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C6),A3))
                   => ( aa(A,B,G,A5) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C6),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,C6) = groups7121269368397514597t_prod(A,B,H,C6) )
                   => ( groups7121269368397514597t_prod(A,B,G,A3) = groups7121269368397514597t_prod(A,B,H,B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3872_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,S3) = groups7121269368397514597t_prod(A,B,G,T4) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3873_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,T4) = groups7121269368397514597t_prod(A,B,G,S3) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3874_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,H,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( groups7121269368397514597t_prod(A,B,G,S3) = groups7121269368397514597t_prod(A,B,H,T4) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3875_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( groups7121269368397514597t_prod(A,B,G,T4) = groups7121269368397514597t_prod(A,B,H,S3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3876_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B4: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),B4),A3)
         => ( aa(set(A),$o,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),B4))),groups7121269368397514597t_prod(A,B,G,B4)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3877_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [I3: A] :
                ( member(A,I3,A3)
               => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3))
                  & aa(B,$o,ord_less(B,aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,ord_less(B,groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3878_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xd: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xd,A3)
           => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xd)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_3879_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xd: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( groups7121269368397514597t_prod(A,B,G,insert(A,Xd,A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xd)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3880_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ca(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2),S3) = $ite(member(A,A2,S3),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),S3),insert(A,A2,bot_bot(set(A)))))),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S3),insert(A,A2,bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_3881_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B4),A3))
                 => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,F2,B3)) )
             => ( ! [A5: A] :
                    ( member(A,A5,A3)
                   => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,A5)) )
               => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,B4)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3882_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => ( groups7121269368397514597t_prod(A,B,F2,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,A2,bot_bot(set(A))))) = $ite(member(A,A2,A3),divide_divide(B,groups7121269368397514597t_prod(A,B,F2,A3),aa(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ) ) ).

% prod_diff1
tff(fact_3883_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Bound: real,I: real] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,ord_less_eq(real,aa(A,real,F2,X)),Bound) )
     => aa(real,$o,ord_less_eq(real,aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),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_3884_ln__series,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),Xd) = suminf(real,aTP_Lamp_cb(real,fun(nat,real),Xd)) ) ) ) ).

% ln_series
tff(fact_3885_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),C2: real,G: fun(A,real),D: real] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,ord_less_eq(real,aa(A,real,F2,X)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X))) )
     => aa(real,$o,ord_less_eq(real,aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xs)),D)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,G),Xs)),zero_zero(real)))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_3886_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,ord_less_eq(nat,aa(A,nat,F2,X)),Bound) )
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F2),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_3887_map__ident,axiom,
    ! [A: $tType,X4: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_cc(A,A)),X4) = X4 ).

% map_ident
tff(fact_3888_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Y: real] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,F2,X)) )
     => aa(real,$o,ord_less_eq(real,Y),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xs)),Y)) ) ).

% listsum_bound
tff(fact_3889_length__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_3890_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),map(B,A,G),Xs) )
    <=> ! [X5: B] :
          ( member(B,X5,aa(list(B),set(B),set2(B),Xs))
         => ( aa(B,A,F2,X5) = aa(B,A,G,X5) ) ) ) ).

% map_eq_conv
tff(fact_3891_map__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Nb: nat,Xd: B] : aa(list(B),list(A),map(B,A,F2),replicate(B,Nb,Xd)) = replicate(A,Nb,aa(B,A,F2,Xd)) ).

% map_replicate
tff(fact_3892_real__nat__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F2),Xs)),C2)) = aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,aTP_Lamp_cd(fun(A,nat),fun(A,real),F2)),Xs)),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% real_nat_list
tff(fact_3893_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F2: fun(A,B)] :
      ( aa(nat,$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,F2),Xs)),Nb) = aa(A,B,F2,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_3894_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_ce(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_3895_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F2),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_3896_map__update,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),K: nat,Y: B] : aa(list(B),list(A),map(B,A,F2),list_update(B,Xs,K,Y)) = list_update(A,aa(list(B),list(A),map(B,A,F2),Xs),K,aa(B,A,F2,Y)) ).

% map_update
tff(fact_3897_list_Omap__ident,axiom,
    ! [A: $tType,T: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_cc(A,A)),T) = T ).

% list.map_ident
tff(fact_3898_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less_eq(multiset(A),M7),N2)
        <=> ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
            | ( M7 = N2 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_3899_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
         => ~ aa(multiset(A),$o,ord_less(multiset(A),N2),M7) ) ) ).

% mset_le_asym
tff(fact_3900_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K6: multiset(A),M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),K6),M7)
         => ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
           => aa(multiset(A),$o,ord_less(multiset(A),K6),N2) ) ) ) ).

% mset_le_trans
tff(fact_3901_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,ord_less(multiset(A),M7),M7) ) ).

% mset_le_irrefl
tff(fact_3902_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
         => ~ aa(multiset(A),$o,ord_less(multiset(A),N2),M7) ) ) ).

% mset_le_not_sym
tff(fact_3903_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,ord_less(multiset(A),M7),M7) ) ).

% mset_le_not_refl
tff(fact_3904_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xd: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xd = Ya )
     => ( ! [Z2: A] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F2,Z2) = aa(A,B,G,Z2) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xd) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong
tff(fact_3905_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xd: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z2: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xd))
         => ( aa(A,B,F2,Z2) = aa(A,B,G,Z2) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xd) = aa(list(A),list(B),map(A,B,G),Xd) ) ) ).

% list.map_cong0
tff(fact_3906_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xd: list(A),Xaa: list(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z2: A,Za: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xd))
         => ( member(A,Za,aa(list(A),set(A),set2(A),Xaa))
           => ( ( aa(A,B,F2,Z2) = aa(A,B,Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F2),Xd) = aa(list(A),list(B),map(A,B,Fa),Xaa) )
       => ( Xd = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_3907_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,G),Xs) ) ) ).

% map_ext
tff(fact_3908_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,A,F2,X) = X ) )
     => ( aa(list(A),list(A),map(A,A,F2),Xs) = Xs ) ) ).

% map_idI
tff(fact_3909_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xs = Ys )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,G),Ys) ) ) ) ).

% map_cong
tff(fact_3910_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F2: fun(A,B)] :
      ( ? [Xs4: list(A)] : Ys = aa(list(A),list(B),map(A,B,F2),Xs4)
    <=> ! [X5: B] :
          ( member(B,X5,aa(list(B),set(B),set2(B),Ys))
         => ? [Xa3: A] : X5 = aa(A,B,F2,Xa3) ) ) ).

% ex_map_conv
tff(fact_3911_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aTP_Lamp_cf(A,fun(B,A),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_3912_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F2: fun(A,B),Xd: A] :
      ( ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,F2,aa(nat,A,nth(A,L),I)) = aa(A,B,F2,Xd) ) )
     => ( aa(list(A),list(B),map(A,B,F2),list_update(A,L,I,Xd)) = aa(list(A),list(B),map(A,B,F2),L) ) ) ).

% map_upd_eq
tff(fact_3913_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(Info,Deg,TreeList,Summary)) = 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))),aa(nat,nat,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'.simps(2)
tff(fact_3914_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Node(Info,Deg,TreeList,Summary)) = 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))),aa(real,real,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.simps(2)
tff(fact_3915_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xd) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( Y != one_one(nat) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( 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,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList2)),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_3916_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xd) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( Y != one_one(real) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( 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,Summary2))),aa(real,real,foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList2)),zero_zero(real))) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_3917_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info,Deg,TreeList,Summary)) = 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))),aa(nat,nat,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'.simps(2)
tff(fact_3918_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xd) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( 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,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList2)),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_3919_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info,Deg,TreeList,Summary)) = 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))),aa(nat,nat,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.simps(2)
tff(fact_3920_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xd) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( 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,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList2)),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_3921_arctan__series,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,real,arctan,Xd) = suminf(real,aTP_Lamp_cg(real,fun(nat,real),Xd)) ) ) ).

% arctan_series
tff(fact_3922_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)) = divide_divide(A,one_one(A),aa(A,A,minus_minus(A,one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3923_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_ch(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_3924_pi__series,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_ci(nat,real)) ).

% pi_series
tff(fact_3925_big__assn__simp,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),L: nat,Xd: vEBT_VEBTi,Xaa: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),Xd)),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xd))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,H,bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xd))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),list_update(vEBT_VEBTi,Tree_is,H,Xd)))) ) ).

% big_assn_simp
tff(fact_3926_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,ord_less_eq(A,L),I)
            & aa(A,$o,ord_less(A,I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_3927_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_3928_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_3929_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_3930_local_Oext,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,Y,bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,Y,bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y)))) ) ).

% local.ext
tff(fact_3931_recomp,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,I,bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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))) ) ).

% recomp
tff(fact_3932_repack,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),Rest: assn,X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),Rest)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,I,bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Rest),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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))) ) ).

% repack
tff(fact_3933_txe,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,Y,bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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))) ) ).

% txe
tff(fact_3934_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = insert(nat,M,bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_3935_tcd,axiom,
    ! [A: $tType,I: nat,TreeList: list(vEBT_VEBT),TreeList4: list(A),Y: vEBT_VEBT,Xd: vEBT_VEBTi,X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(list(A),nat,size_size(list(A)),TreeList4) )
       => 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),Xd)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xd)))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,I,bot_bot(set(nat)))),vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y),list_update(vEBT_VEBTi,Tree_is,I,Xd))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xd))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y)),list_update(vEBT_VEBTi,Tree_is,I,Xd)))) ) ) ).

% tcd
tff(fact_3936_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),Xaa: vEBT_VEBT,L: nat,Xd: vEBT_VEBTi,Xba: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( Xaa = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
       => 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),Xd)),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xd))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),insert(nat,H,bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xd))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H,Xaa)),list_update(vEBT_VEBTi,Tree_is,H,Xd)))) ) ) ).

% big_assn_simp'
tff(fact_3937_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_3938_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D) )
            <=> ( ( A2 = C2 )
                & ( B2 = D ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_3939_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D) )
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less(A,C2),D)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_3940_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D) )
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less(A,C2),D)
             => ( B2 = D ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_3941_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_3942_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_3943_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_3944_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_3945_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,ord_less(nat,M4),Nb)
         => aa(nat,$o,P,M4) )
    <=> ! [X5: nat] :
          ( member(nat,X5,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X5) ) ) ).

% all_nat_less_eq
tff(fact_3946_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M4: nat] :
          ( aa(nat,$o,ord_less(nat,M4),Nb)
          & aa(nat,$o,P,M4) )
    <=> ? [X5: nat] :
          ( member(nat,X5,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X5) ) ) ).

% ex_nat_less_eq
tff(fact_3947_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_3948_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_3949_pi__not__less__zero,axiom,
    ~ aa(real,$o,ord_less(real,pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_3950_pi__gt__zero,axiom,
    aa(real,$o,ord_less(real,zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_3951_pi__ge__zero,axiom,
    aa(real,$o,ord_less_eq(real,zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_3952_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B),F3: 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)),F3) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi)),F3) ) ) ).

% listI_assn_conv'
tff(fact_3953_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_bd(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_3954_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_be(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_3955_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C2: A,B2: A,D: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D )
           => ( ! [X: A] :
                  ( aa(A,$o,ord_less_eq(A,C2),X)
                 => ( aa(A,$o,ord_less(A,X),D)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( groups7121269368397514597t_prod(A,B,G,set_or7035219750837199246ssThan(A,A2,B2)) = groups7121269368397514597t_prod(A,B,H,set_or7035219750837199246ssThan(A,C2,D)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3956_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ( aa(nat,$o,ord_less_eq(nat,Nb),P3)
           => ( 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,P3))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3957_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = insert(nat,Nb,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_3958_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite2(nat),N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_3959_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I6: set(nat),I7: set(nat),Xs: list(A),Xs3: list(A),Xsi: list(B),Xsi2: list(B),A3: fun(A,fun(B,assn)),A6: fun(A,fun(B,assn))] :
      ( ( I6 = I7 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs3) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
         => ( ! [I3: nat] :
                ( member(nat,I3,I6)
               => ( aa(nat,$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,Xs3),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),A6,aa(nat,A,nth(A,Xs3),I3)),aa(nat,B,nth(B,Xsi2),I3)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I7,A6,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3960_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I6: set(nat),I7: set(nat),A3: fun(A,fun(B,assn)),A6: fun(A,fun(B,assn)),Xs: list(A),Xs3: list(A),Xsi: list(B),Xsi2: list(B)] :
      ( ( I6 = I7 )
     => ( ( A3 = A6 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs3) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
           => ( ! [I3: nat] :
                  ( member(nat,I3,I6)
                 => ( aa(nat,$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,Xs3),I3) )
                        & ( aa(nat,B,nth(B,Xsi),I3) = aa(nat,B,nth(B,Xsi2),I3) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I7,A6,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3961_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less_eq(A,B2),D) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3962_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less_eq(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less(A,B2),D) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3963_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I6: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X15: A,Xsi: list(B),X2: B] :
      ( ~ member(nat,I,I6)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I6,A3,list_update(A,Xs,I,X15),list_update(B,Xsi,I,X2)) = vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_3964_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_3965_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_3966_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$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_3967_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_3968_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_3969_pi__ge__two,axiom,
    aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_3970_pi__half__neq__two,axiom,
    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_3971_atLeastLessThanSuc,axiom,
    ! [M: nat,Nb: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,ord_less_eq(nat,M),Nb),insert(nat,Nb,set_or7035219750837199246ssThan(nat,M,Nb)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_3972_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_cj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M),set_or7035219750837199246ssThan(nat,Nb,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_3973_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_ck(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_cm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% prod.nested_swap
tff(fact_3974_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_3975_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_3976_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_3977_pi__half__neq__zero,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_3978_pi__half__less__two,axiom,
    aa(real,$o,ord_less(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_3979_pi__half__le__two,axiom,
    aa(real,$o,ord_less_eq(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_3980_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,ord_less_eq(nat,M),pred_numeral(K)),insert(nat,pred_numeral(K),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_3981_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_3982_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_bf(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_3983_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_bi(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_3984_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_3985_pi__half__gt__zero,axiom,
    aa(real,$o,ord_less(real,zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_3986_pi__half__ge__zero,axiom,
    aa(real,$o,ord_less_eq(real,zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_ge_zero
tff(fact_3987_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_3988_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,ord_less(real,aa(real,real,arctan,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arctan_ubound
tff(fact_3989_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% arctan_one
tff(fact_3990_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I6: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ~ member(nat,I,I6)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,insert(nat,I,I6),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,I6,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_3991_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I6: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X15: A,Xsi: list(B),X2: B] :
      ( ~ member(nat,I,I6)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,insert(nat,I,I6),A3,list_update(A,Xs,I,X15),list_update(B,Xsi,I,X2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,X15),X2)),vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_3992_minus__pi__half__less__zero,axiom,
    aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_3993_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$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_3994_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,ord_less(real,aa(real,real,arctan,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_3995_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y)) ).

% arctan_lbound
tff(fact_3996_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$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_cn(nat,fun(nat,fun(nat,A)),K),Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_3997_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I6: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( member(nat,I,I6)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I6,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),I6),insert(nat,I,bot_bot(set(nat)))),A3,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3998_machin__Euler,axiom,
    aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2)))),aa(real,real,arctan,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,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))))))))))) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% machin_Euler
tff(fact_3999_machin,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(real,real,arctan,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2))))))),aa(real,real,arctan,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_4000_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I6: set(nat),F3: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I6),insert(nat,I,bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I6)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi)),F3),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4001_listI__assn__reinsert__upd,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xd: A,Xi2: B,I6: set(nat),I: nat,Xs: list(A),Xsi: list(B),F3: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xd),Xi2)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I6),insert(nat,I,bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I6)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I6,A3,list_update(A,Xs,I,Xd),list_update(B,Xsi,I,Xi2))),F3),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4002_arctan__inverse,axiom,
    ! [Xd: real] :
      ( ( Xd != zero_zero(real) )
     => ( aa(real,real,arctan,divide_divide(real,one_one(real),Xd)) = aa(real,real,minus_minus(real,divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Xd)),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,real,arctan,Xd)) ) ) ).

% arctan_inverse
tff(fact_4003_sin__cos__npi,axiom,
    ! [Nb: nat] : sin(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_4004_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),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_4005_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,ord_less_eq(num,M),Nb),aa(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_4006_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q2),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_4007_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_4008_sin__pi,axiom,
    sin(real,pi) = zero_zero(real) ).

% sin_pi
tff(fact_4009_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,real_Vector_of_real(A,pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_4010_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(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_4011_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_4012_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_4013_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_4014_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_4015_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,bit0(Nb)) = aa(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_4016_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),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_4017_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_4018_sin__pi__half,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_4019_sin__periodic,axiom,
    ! [Xd: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = sin(real,Xd) ).

% sin_periodic
tff(fact_4020_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(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_4021_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_4022_sin__2pi__minus,axiom,
    ! [Xd: 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)),Xd)) = aa(real,real,uminus_uminus(real),sin(real,Xd)) ).

% sin_2pi_minus
tff(fact_4023_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_4024_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(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_4025_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_4026_sin__x__le__x,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,sin(real,Xd)),Xd) ) ).

% sin_x_le_x
tff(fact_4027_sin__le__one,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,sin(real,Xd)),one_one(real)) ).

% sin_le_one
tff(fact_4028_word__to__1__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),one_one(word(A))) = insert(word(A),zero_zero(word(A)),bot_bot(set(word(A)))) ) ) ).

% word_to_1_set
tff(fact_4029_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),$o,finite_finite2(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_4030_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_4031_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,Xd: 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,Xd))) = 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)),Xd))) ) ).

% sin_int_times_real
tff(fact_4032_sin__gt__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),pi)
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,Xd)) ) ) ).

% sin_gt_zero
tff(fact_4033_sin__x__ge__neg__x,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),Xd)),sin(real,Xd)) ) ).

% sin_x_ge_neg_x
tff(fact_4034_sin__ge__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),pi)
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,Xd)) ) ) ).

% sin_ge_zero
tff(fact_4035_sin__ge__minus__one,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),sin(real,Xd)) ).

% sin_ge_minus_one
tff(fact_4036_abs__sin__le__one,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,sin(real,Xd))),one_one(real)) ).

% abs_sin_le_one
tff(fact_4037_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_4038_sin__eq__0__pi,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),pi)
       => ( ( sin(real,Xd) = zero_zero(real) )
         => ( Xd = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_4039_divmod__int__def,axiom,
    ! [M: num,Nb: num] : unique8689654367752047608divmod(int,M,Nb) = aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb))) ).

% divmod_int_def
tff(fact_4040_sin__zero__pi__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),pi)
     => ( ( sin(real,Xd) = zero_zero(real) )
      <=> ( Xd = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_4041_sin__zero__iff__int2,axiom,
    ! [Xd: real] :
      ( ( sin(real,Xd) = zero_zero(real) )
    <=> ? [I2: int] : Xd = 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_4042_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_4043_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_4044_sin__gt__zero__02,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,Xd)) ) ) ).

% sin_gt_zero_02
tff(fact_4045_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] : unique8689654367752047608divmod(A,M,Nb) = aa(A,product_prod(A,A),product_Pair(A,A,divide_divide(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Nb))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Nb))) ) ).

% divmod_def
tff(fact_4046_divmod_H__nat__def,axiom,
    ! [M: num,Nb: num] : unique8689654367752047608divmod(nat,M,Nb) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,divide_divide(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Nb))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Nb))) ).

% divmod'_nat_def
tff(fact_4047_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_4048_sin__gt__zero2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,Xd)) ) ) ).

% sin_gt_zero2
tff(fact_4049_sin__lt__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,pi),Xd)
     => ( aa(real,$o,ord_less(real,Xd),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,Xd)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_4050_sin__30,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_30
tff(fact_4051_sin__inj__pi,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,Xd) = sin(real,Y) )
             => ( Xd = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_4052_sin__mono__le__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,sin(real,Xd)),sin(real,Y))
            <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_4053_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,sin(real,Y)),sin(real,Xd)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_4054_sin__le__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,pi),Xd)
     => ( aa(real,$o,ord_less(real,Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,ord_less_eq(real,sin(real,Xd)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_4055_sin__less__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),zero_zero(real))
       => aa(real,$o,ord_less(real,sin(real,Xd)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_4056_sin__monotone__2pi,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less(real,Y),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less(real,sin(real,Y)),sin(real,Xd)) ) ) ) ).

% sin_monotone_2pi
tff(fact_4057_sin__mono__less__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less(real,sin(real,Xd)),sin(real,Y))
            <=> aa(real,$o,ord_less(real,Xd),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_4058_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X)
            & aa(real,$o,ord_less_eq(real,X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X) = Y )
            & ! [Y3: real] :
                ( ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y3)
                  & aa(real,$o,ord_less_eq(real,Y3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y3) = Y ) )
               => ( Y3 = X ) ) ) ) ) ).

% sin_total
tff(fact_4059_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa(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,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_4060_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),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_4061_sin__zero__iff__int,axiom,
    ! [Xd: real] :
      ( ( sin(real,Xd) = zero_zero(real) )
    <=> ? [I2: int] :
          ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I2)
          & ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_4062_sin__zero__lemma,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( ( sin(real,Xd) = zero_zero(real) )
       => ? [N: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_4063_sin__zero__iff,axiom,
    ! [Xd: real] :
      ( ( sin(real,Xd) = zero_zero(real) )
    <=> ( ? [N6: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N6: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xd = 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_4064_one__div__minus__numeral,axiom,
    ! [Nb: num] : 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_4065_minus__one__div__numeral,axiom,
    ! [Nb: num] : 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_4066_htt__cons__rule,axiom,
    ! [A: $tType,P2: assn,C2: heap_Time_Heap(A),Q3: fun(A,assn),T5: nat,P: assn,Q: fun(A,assn),T: nat] :
      ( time_htt(A,P2,C2,Q3,T5)
     => ( entails(P,P2)
       => ( ! [X: A] : entails(aa(A,assn,Q3,X),aa(A,assn,Q,X))
         => ( aa(nat,$o,ord_less_eq(nat,T5),T)
           => time_htt(A,P,C2,Q,T) ) ) ) ) ).

% htt_cons_rule
tff(fact_4067_summable__arctan__series,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => summable(real,aTP_Lamp_cg(real,fun(nat,real),Xd)) ) ).

% summable_arctan_series
tff(fact_4068_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_co(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_4069_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_cp(nat,A)) ) ).

% summable_zero
tff(fact_4070_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_4071_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cr(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_4072_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cs(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_4073_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A3)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ct(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_4074_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cu(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_4075_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_4076_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_4077_minus__numeral__div__numeral,axiom,
    ! [M: num,Nb: num] : 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_4078_numeral__div__minus__numeral,axiom,
    ! [M: num,Nb: num] : 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_4079_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_cv(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_4080_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_4081_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_4082_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_4083_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cy(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_4084_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_4085_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cs(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_4086_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Xd: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_da(fun(nat,A),fun(A,fun(nat,A)),F2),Xd))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xd))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_4087_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N2)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_4088_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cr(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_4089_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_4090_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_4091_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_cy(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F2)) ) ) ) ).

% suminf_mult
tff(fact_4092_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F2)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_4093_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_cs(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = divide_divide(A,suminf(A,F2),C2) ) ) ) ).

% suminf_divide
tff(fact_4094_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_4095_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N6: nat] : aa(nat,A,F2,N6) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_4096_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_pos
tff(fact_4097_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_dc(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_4098_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_4099_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_da(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_4100_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_4101_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dh(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_da(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_4102_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,I))
             => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_4103_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F2))
            <=> ? [I2: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,I2)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_4104_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Xd: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),Xd))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xd))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_4105_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),F2)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_4106_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_4107_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),Xd)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_4108_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F2,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_4109_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_4110_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
         => ( summable(A,F2)
           => ? [N8: nat] :
              ! [N9: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N8),N9)
               => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),N9)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_4111_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,I3)),one_one(real))
     => ( ! [I3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,I3))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Z)
         => ( aa(real,$o,ord_less(real,Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_di(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_4112_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M7: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),R2)
         => ( aa(real,$o,ord_less(real,R2),R0)
           => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N))),M7)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_dj(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_4113_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N2: nat,F2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N2),N)
               => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N)))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_4114_norm__pre__pure__iff__htt,axiom,
    ! [A: $tType,P: assn,B2: $o,F2: heap_Time_Heap(A),Q: fun(A,assn),T: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F2,Q,T)
    <=> ( (B2)
       => time_htt(A,P,F2,Q,T) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_4115_norm__pre__pure__iff__htt_H,axiom,
    ! [A: $tType,B2: $o,P: assn,F2: heap_Time_Heap(A),Q: fun(A,assn),T: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((B2))),P),F2,Q,T)
    <=> ( (B2)
       => time_htt(A,P,F2,Q,T) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_4116_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R2: int] :
      ( aa(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),product_Pair(int,int,Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),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),product_Pair(int,int,Q2),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4117_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),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),product_Pair(int,int,Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4118_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),product_Pair(A,B,aa(nat,A,nth(A,Xs),divide_divide(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_4119_cos__pi__eq__zero,axiom,
    ! [M: nat] : cos(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_4120_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_4121_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_4122_cos__pi,axiom,
    cos(real,pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_4123_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: 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,Xd)),cos(A,Xd))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xd)),sin(A,Xd))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_4124_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_4125_cos__pi__half,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_4126_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_4127_cos__periodic,axiom,
    ! [Xd: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = cos(real,Xd) ).

% cos_periodic
tff(fact_4128_cos__2pi__minus,axiom,
    ! [Xd: 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)),Xd)) = cos(real,Xd) ).

% cos_2pi_minus
tff(fact_4129_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_4130_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_4131_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: 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,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_4132_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: 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,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_4133_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(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_4134_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_4135_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_4136_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(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_4137_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_4138_unique__remainder,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q6),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_4139_unique__quotient,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q6),R4))
       => ( Q2 = Q6 ) ) ) ).

% unique_quotient
tff(fact_4140_cos__le__one,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,cos(real,Xd)),one_one(real)) ).

% cos_le_one
tff(fact_4141_polar__Ex,axiom,
    ! [Xd: real,Y: real] :
    ? [R3: real,A5: real] :
      ( ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),R3),cos(real,A5)) )
      & ( Y = aa(real,real,aa(real,fun(real,real),times_times(real),R3),sin(real,A5)) ) ) ).

% polar_Ex
tff(fact_4142_cos__arctan__not__zero,axiom,
    ! [Xd: real] : cos(real,aa(real,real,arctan,Xd)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_4143_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,Xd: 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,Xd))) = 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)),Xd))) ) ).

% cos_int_times_real
tff(fact_4144_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( cos(A,Xd) = one_one(A) )
         => ( sin(A,Xd) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_4145_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),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,Xd)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xd)),sin(A,Y))) ) ).

% sin_add
tff(fact_4146_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : sin(A,aa(A,A,minus_minus(A,Xd),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xd)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xd)),sin(A,Y))) ) ).

% sin_diff
tff(fact_4147_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),pi)
         => aa(real,$o,ord_less_eq(real,cos(real,Xd)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_4148_cos__mono__le__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( aa(real,$o,ord_less_eq(real,cos(real,Xd)),cos(real,Y))
            <=> aa(real,$o,ord_less_eq(real,Y),Xd) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_4149_cos__inj__pi,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( ( cos(real,Xd) = cos(real,Y) )
             => ( Xd = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_4150_cos__ge__minus__one,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),cos(real,Xd)) ).

% cos_ge_minus_one
tff(fact_4151_abs__cos__le__one,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,cos(real,Xd))),one_one(real)) ).

% abs_cos_le_one
tff(fact_4152_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_4153_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( modulo_modulo(int,K,L) = R2 ) ) ).

% mod_int_unique
tff(fact_4154_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( divide_divide(int,K,L) = Q2 ) ) ).

% div_int_unique
tff(fact_4155_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xd)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xd)),sin(A,Y))) ) ).

% cos_add
tff(fact_4156_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : cos(A,aa(A,A,minus_minus(A,Xd),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,Xd)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xd)),sin(A,Y))) ) ).

% cos_diff
tff(fact_4157_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( sin(A,Xd) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,Xd)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_4158_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_4159_cos__monotone__0__pi,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less(real,Y),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),pi)
         => aa(real,$o,ord_less(real,cos(real,Xd)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_4160_cos__mono__less__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( aa(real,$o,ord_less(real,cos(real,Xd)),cos(real,Y))
            <=> aa(real,$o,ord_less(real,Y),Xd) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_4161_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
         => aa(real,$o,ord_less_eq(real,cos(real,Y)),cos(real,Xd)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_4162_sin__zero__abs__cos__one,axiom,
    ! [Xd: real] :
      ( ( sin(real,Xd) = zero_zero(real) )
     => ( abs_abs(real,cos(real,Xd)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_4163_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)) = 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,Xd))),cos(A,Xd)) ) ).

% sin_double
tff(fact_4164_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),product_Pair(int,int,Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4165_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,K,L)),modulo_modulo(int,K,L))) ).

% eucl_rel_int
tff(fact_4166_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_4167_cos__is__zero,axiom,
    ? [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
      & aa(real,$o,ord_less_eq(real,X),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X) = zero_zero(real) )
      & ! [Y3: real] :
          ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y3)
            & aa(real,$o,ord_less_eq(real,Y3),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y3) = zero_zero(real) ) )
         => ( Y3 = X ) ) ) ).

% cos_is_zero
tff(fact_4168_cos__two__le__zero,axiom,
    aa(real,$o,ord_less_eq(real,cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_4169_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,ord_less(real,Y),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
         => aa(real,$o,ord_less(real,cos(real,Y)),cos(real,Xd)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_4170_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
            & aa(real,$o,ord_less_eq(real,X),pi)
            & ( cos(real,X) = Y )
            & ! [Y3: real] :
                ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y3)
                  & aa(real,$o,ord_less_eq(real,Y3),pi)
                  & ( cos(real,Y3) = Y ) )
               => ( Y3 = X ) ) ) ) ) ).

% cos_total
tff(fact_4171_sincos__principal__value,axiom,
    ! [Xd: real] :
    ? [Y2: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Y2)
      & aa(real,$o,ord_less_eq(real,Y2),pi)
      & ( sin(real,Y2) = sin(real,Xd) )
      & ( cos(real,Y2) = cos(real,Xd) ) ) ).

% sincos_principal_value
tff(fact_4172_sin__cos__le1,axiom,
    ! [Xd: real,Y: real] : aa(real,$o,ord_less_eq(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,Xd)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,Xd)),cos(real,Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_4173_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)) = 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_4174_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,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,divide_divide(A,aa(A,A,minus_minus(A,W),Z),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_4175_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xd)),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,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sin_squared_eq
tff(fact_4176_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xd)),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,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_squared_eq
tff(fact_4177_cos__double__less__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),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))),Xd))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_4178_cos__gt__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),cos(real,Xd)) ) ) ).

% cos_gt_zero
tff(fact_4179_cos__60,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_60
tff(fact_4180_cos__one__2pi__int,axiom,
    ! [Xd: real] :
      ( ( cos(real,Xd) = one_one(real) )
    <=> ? [X5: int] : Xd = 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),X5)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_4181_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_4182_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: 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))),Xd)) = 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,Xd)),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,Xd))) ) ).

% cos_treble_cos
tff(fact_4183_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)) = 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_4184_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)) = 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_4185_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)) = 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_4186_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,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,divide_divide(A,aa(A,A,minus_minus(A,W),Z),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_4187_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,divide_divide(A,aa(A,A,minus_minus(A,W),Z),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(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_4188_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,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,divide_divide(A,aa(A,A,minus_minus(A,Z),W),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_4189_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_4190_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : sin(A,Xd) = cos(A,aa(A,A,minus_minus(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2)))),Xd)) ) ).

% sin_cos_eq
tff(fact_4191_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : cos(A,Xd) = sin(A,aa(A,A,minus_minus(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2)))),Xd)) ) ).

% cos_sin_eq
tff(fact_4192_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,
            aa(int,product_prod(int,int),
              product_Pair(int,int,
                $ite(R2 = zero_zero(int),aa(int,int,uminus_uminus(int),Q2),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_4193_cos__gt__zero__pi,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),cos(real,Xd)) ) ) ).

% cos_gt_zero_pi
tff(fact_4194_cos__ge__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),cos(real,Xd)) ) ) ).

% cos_ge_zero
tff(fact_4195_cos__one__2pi,axiom,
    ! [Xd: real] :
      ( ( cos(real,Xd) = one_one(real) )
    <=> ( ? [X5: nat] : Xd = 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),X5)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X5: nat] : Xd = 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),X5)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_4196_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_4197_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(A,A,uminus_uminus(A),sin(A,Xd)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% minus_sin_cos_eq
tff(fact_4198_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R2) )
        & $ite(
            aa(int,$o,ord_less(int,zero_zero(int)),L),
            ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
            & aa(int,$o,ord_less(int,R2),L) ),
            $ite(
              aa(int,$o,ord_less(int,L),zero_zero(int)),
              ( aa(int,$o,ord_less(int,L),R2)
              & aa(int,$o,ord_less_eq(int,R2),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4199_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( sgn_sgn(int,R2) = sgn_sgn(int,L) )
     => ( aa(int,$o,ord_less(int,abs_abs(int,R2)),abs_abs(int,L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_4200_sincos__total__pi,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
       => ? [T6: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
            & aa(real,$o,ord_less_eq(real,T6),pi)
            & ( Xd = cos(real,T6) )
            & ( Y = sin(real,T6) ) ) ) ) ).

% sincos_total_pi
tff(fact_4201_sin__expansion__lemma,axiom,
    ! [Xd: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),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),Xd),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_4202_cos__zero__iff__int,axiom,
    ! [Xd: real] :
      ( ( cos(real,Xd) = zero_zero(real) )
    <=> ? [I2: int] :
          ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I2)
          & ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_4203_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K3) ) )
        | ? [L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q5),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q5),R5) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L4) )
            & aa(int,$o,ord_less(int,abs_abs(int,R5)),abs_abs(int,L4))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_4204_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),product_Pair(int,int,zero_zero(int)),A1) ) )
       => ( ! [Q4: int] :
              ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A22) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),R3) )
               => ( ( sgn_sgn(int,R3) = sgn_sgn(int,A22) )
                 => ( aa(int,$o,ord_less(int,abs_abs(int,R3)),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),Q4),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_4205_cos__zero__lemma,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( ( cos(real,Xd) = zero_zero(real) )
       => ? [N: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_4206_cos__zero__iff,axiom,
    ! [Xd: real] :
      ( ( cos(real,Xd) = zero_zero(real) )
    <=> ( ? [N6: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xd = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N6: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xd = 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_4207_cos__expansion__lemma,axiom,
    ! [Xd: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),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),Xd),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_4208_sincos__total__pi__half,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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) )
         => ? [T6: real] :
              ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
              & aa(real,$o,ord_less_eq(real,T6),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xd = cos(real,T6) )
              & ( Y = sin(real,T6) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_4209_sincos__total__2pi__le,axiom,
    ! [Xd: 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),Xd),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) )
     => ? [T6: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
          & aa(real,$o,ord_less_eq(real,T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( Xd = cos(real,T6) )
          & ( Y = sin(real,T6) ) ) ) ).

% sincos_total_2pi_le
tff(fact_4210_sincos__total__2pi,axiom,
    ! [Xd: 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),Xd),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) )
     => ~ ! [T6: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
           => ( aa(real,$o,ord_less(real,T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xd = cos(real,T6) )
               => ( Y != sin(real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_4211_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),Xd)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_4212__C11_C,axiom,
    ! [Xaa: array(vEBT_VEBTi),Xd: list(vEBT_VEBTi)] : hoare_hoare_triple(vEBT_VEBTi,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,Xaa),Xd)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,replicate(vEBT_VEBT,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))),divide_divide(nat,na,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,na,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),Xd)),vEBT_V739175172307565963ildupi(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,na)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBTi),fun(vEBT_VEBTi,assn),aTP_Lamp_dk(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),fun(vEBT_VEBTi,assn)),Xaa),Xd)) ).

% "11"
tff(fact_4213_sgn__integer__code,axiom,
    ! [K: 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_4214_infinite__int__iff__unbounded,axiom,
    ! [S3: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S3)
    <=> ! [M4: int] :
        ? [N6: int] :
          ( aa(int,$o,ord_less(int,M4),abs_abs(int,N6))
          & member(int,N6,S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_4215_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_4216_tan__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tan(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tan_zero
tff(fact_4217__C3_OIH_C_I4_J,axiom,
    ! [Xd: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na)))
     => ( ( Xd = divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
       => hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(aa(nat,nat,suc,Xd)),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa(nat,nat,suc,Xd)))) ) ) ).

% "3.IH"(4)
tff(fact_4218__C3_OIH_C_I3_J,axiom,
    ! [Xd: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na)))
     => ( ( Xd = divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
       => hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Xd),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Xd))) ) ) ).

% "3.IH"(3)
tff(fact_4219__C3_OIH_C_I1_J,axiom,
    ! [Xd: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na)))
     => ( ( Xd = divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
       => hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Xd),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Xd))) ) ) ).

% "3.IH"(1)
tff(fact_4220_tan__npi,axiom,
    ! [Nb: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_4221_tan__periodic__n,axiom,
    ! [Xd: real,Nb: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),Nb)),pi))) = aa(real,real,tan(real),Xd) ).

% tan_periodic_n
tff(fact_4222_tan__periodic__nat,axiom,
    ! [Xd: real,Nb: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi))) = aa(real,real,tan(real),Xd) ).

% tan_periodic_nat
tff(fact_4223_tan__periodic__int,axiom,
    ! [Xd: real,I: int] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I)),pi))) = aa(real,real,tan(real),Xd) ).

% tan_periodic_int
tff(fact_4224_tan__periodic,axiom,
    ! [Xd: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,tan(real),Xd) ).

% tan_periodic
tff(fact_4225_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_dl(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule_left
tff(fact_4226_fi__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ps: assn,F3: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F3))
       => hoare_hoare_triple(A,Ps,C2,aa(assn,fun(A,assn),aTP_Lamp_dm(fun(A,assn),fun(assn,fun(A,assn)),Q),F3)) ) ) ).

% fi_rule
tff(fact_4227_return__cons__rule,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xd: A] :
      ( entails(P,aa(A,assn,Q,Xd))
     => hoare_hoare_triple(A,P,heap_Time_return(A,Xd),Q) ) ).

% return_cons_rule
tff(fact_4228_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_4229_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_4230_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_4231_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_4232_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_4233_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_4234_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_4235_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_4236_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: A] : aa(A,A,tan(A),X4) = divide_divide(A,sin(A,X4),cos(A,X4)) ) ).

% tan_def
tff(fact_4237_tan__45,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = one_one(real) ).

% tan_45
tff(fact_4238_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),X)
          & aa(real,$o,ord_less(real,X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,ord_less(real,Y),aa(real,real,tan(real),X)) ) ) ).

% lemma_tan_total
tff(fact_4239_tan__gt__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,tan(real),Xd)) ) ) ).

% tan_gt_zero
tff(fact_4240_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X)
      & aa(real,$o,ord_less(real,X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y ) ) ).

% lemma_tan_total1
tff(fact_4241_tan__mono__lt__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less(real,aa(real,real,tan(real),Xd)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,ord_less(real,Xd),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_4242_tan__monotone_H,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
         => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less(real,Y),Xd)
            <=> aa(real,$o,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xd)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_4243_tan__monotone,axiom,
    ! [Y: real,Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less(real,Y),Xd)
       => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xd)) ) ) ) ).

% tan_monotone
tff(fact_4244_tan__total,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X)
      & aa(real,$o,ord_less(real,X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y )
      & ! [Y3: real] :
          ( ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y3)
            & aa(real,$o,ord_less(real,Y3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y3) = Y ) )
         => ( Y3 = X ) ) ) ).

% tan_total
tff(fact_4245_tan__minus__45,axiom,
    aa(real,real,tan(real),aa(real,real,uminus_uminus(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_4246_tan__inverse,axiom,
    ! [Y: real] : divide_divide(real,one_one(real),aa(real,real,tan(real),Y)) = aa(real,real,tan(real),aa(real,real,minus_minus(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),Y)) ).

% tan_inverse
tff(fact_4247_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xd)),aa(A,A,tan(A),Y)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xd)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_4248_infinite__nat__iff__unbounded,axiom,
    ! [S3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S3)
    <=> ! [M4: nat] :
        ? [N6: nat] :
          ( aa(nat,$o,ord_less(nat,M4),N6)
          & member(nat,N6,S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_4249_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set(nat)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,ord_less(nat,K),M2)
         => ? [N9: nat] :
              ( aa(nat,$o,ord_less(nat,M2),N9)
              & member(nat,N9,S3) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S3) ) ).

% unbounded_k_infinite
tff(fact_4250_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
          & aa(real,$o,ord_less(real,X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( aa(real,real,tan(real),X) = Y ) ) ) ).

% tan_total_pos
tff(fact_4251_tan__pos__pi2__le,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,tan(real),Xd)) ) ) ).

% tan_pos_pi2_le
tff(fact_4252_tan__less__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),zero_zero(real))
       => aa(real,$o,ord_less(real,aa(real,real,tan(real),Xd)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_4253_tan__mono__le,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),Y)
       => ( aa(real,$o,ord_less(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,aa(real,real,tan(real),Xd)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_4254_tan__mono__le__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,aa(real,real,tan(real),Xd)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_4255_tan__bound__pi2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,tan(real),Xd))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_4256_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,ord_less(real,aa(real,real,arctan,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_4257_arctan__tan,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),Xd)) = Xd ) ) ) ).

% arctan_tan
tff(fact_4258_arctan__unique,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),Xd) = Y )
         => ( aa(real,real,arctan,Y) = Xd ) ) ) ) ).

% arctan_unique
tff(fact_4259_abs__integer__code,axiom,
    ! [K: code_integer] :
      abs_abs(code_integer,K) = $ite(aa(code_integer,$o,ord_less(code_integer,K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_4260_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_4261_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_4262_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_4263_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xd)),aa(A,A,tan(A),Y)),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xd)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_4264_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,Xd),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,minus_minus(A,Xd),Y)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,tan(A),Xd)),aa(A,A,tan(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xd)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_4265_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xd)),aa(A,A,tan(A),Y))) = divide_divide(A,cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xd)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_4266_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_4267_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_4268_tan__total__pi4,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),one_one(real))
     => ? [Z2: real] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z2)
          & aa(real,$o,ord_less(real,Z2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z2) = Xd ) ) ) ).

% tan_total_pi4
tff(fact_4269_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I6: set(nat),F3: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I6),insert(nat,I,bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I6)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi)),F3),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4270_listI__assn__reinsert__upd_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xd: A,Xi2: B,I6: set(nat),I: nat,Xs: list(A),Xsi: list(B),F3: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xd),Xi2)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I6),insert(nat,I,bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I6)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I6,A3,list_update(A,Xs,I,Xd),list_update(B,Xsi,I,Xi2))),F3),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4271_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(A,A,tan(A),Xd) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)),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))),Xd))),one_one(A))) ) ).

% tan_half
tff(fact_4272_rule__at__index,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),F3: assn,I: nat,C2: heap_Time_Heap(C),Q3: fun(C,assn),F5: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi)),F3))
     => ( aa(nat,$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))),insert(nat,I,bot_bot(set(nat)))),A3,Xs,Xsi))),F3),C2,Q3)
         => ( ! [R3: C] : entails(aa(C,assn,Q3,R3),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))),insert(nat,I,bot_bot(set(nat)))),A3,Xs,Xsi))),aa(C,assn,F5,R3)))
           => 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_dn(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),A3),Xs),Xsi),F5)) ) ) ) ) ).

% rule_at_index
tff(fact_4273_repli__emp,axiom,
    ! [A: $tType,B: $tType,Xd: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Nb: nat] :
      ( hoare_hoare_triple(A,one_one(assn),Xd,aa(B,fun(A,assn),A3,Y))
     => hoare_hoare_triple(list(A),one_one(assn),vEBT_VEBT_replicatei(A,Nb,Xd),vEBT_List_list_assn(B,A,A3,replicate(B,Nb,Y))) ) ).

% repli_emp
tff(fact_4274_repli__cons__repl,axiom,
    ! [B: $tType,A: $tType,Q: assn,Xd: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Nb: nat] :
      ( hoare_hoare_triple(A,Q,Xd,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_do(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,Xd),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_dp(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Q),A3),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_4275_T__vebt__buildupi,axiom,
    ! [Nb: nat,H: heap_ext(product_unit)] : aa(nat,$o,ord_less_eq(nat,time_time(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Nb),H)),vEBT_V441764108873111860ildupi(Nb)) ).

% T_vebt_buildupi
tff(fact_4276_norm__pre__pure__iff__sng,axiom,
    ! [A: $tType,B2: $o,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,pure_assn((B2)),F2,Q)
    <=> ( (B2)
       => hoare_hoare_triple(A,one_one(assn),F2,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_4277_highi__h,axiom,
    ! [Xd: nat,Nb: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_highi(Xd,Nb),aa(nat,fun(nat,assn),aTP_Lamp_ar(nat,fun(nat,fun(nat,assn)),Xd),Nb)) ).

% highi_h
tff(fact_4278_lowi__h,axiom,
    ! [Xd: nat,Nb: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_lowi(Xd,Nb),aa(nat,fun(nat,assn),aTP_Lamp_aq(nat,fun(nat,fun(nat,assn)),Xd),Nb)) ).

% lowi_h
tff(fact_4279_time__replicate,axiom,
    ! [A: $tType,Xd: heap_Time_Heap(A),C2: nat,Nb: nat,H: heap_ext(product_unit)] :
      ( ! [H3: heap_ext(product_unit)] : aa(nat,$o,ord_less_eq(nat,time_time(A,Xd,H3)),C2)
     => aa(nat,$o,ord_less_eq(nat,time_time(list(A),vEBT_VEBT_replicatei(A,Nb,Xd),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_4280_norm__pre__pure__iff,axiom,
    ! [A: $tType,P: assn,B2: $o,F2: 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))),F2,Q)
    <=> ( (B2)
       => hoare_hoare_triple(A,P,F2,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_4281_time__return,axiom,
    ! [A: $tType,Xd: A,H: heap_ext(product_unit)] : time_time(A,heap_Time_return(A,Xd),H) = one_one(nat) ).

% time_return
tff(fact_4282_cons__rule,axiom,
    ! [A: $tType,P: assn,P2: assn,Q: fun(A,assn),Q3: fun(A,assn),C2: heap_Time_Heap(A)] :
      ( entails(P,P2)
     => ( ! [X: A] : entails(aa(A,assn,Q,X),aa(A,assn,Q3,X))
       => ( hoare_hoare_triple(A,P2,C2,Q)
         => hoare_hoare_triple(A,P,C2,Q3) ) ) ) ).

% cons_rule
tff(fact_4283_cons__post__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Q3: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( ! [X: A] : entails(aa(A,assn,Q,X),aa(A,assn,Q3,X))
       => hoare_hoare_triple(A,P,C2,Q3) ) ) ).

% cons_post_rule
tff(fact_4284_return__wp__rule,axiom,
    ! [A: $tType,Q: fun(A,assn),Xd: A] : hoare_hoare_triple(A,aa(A,assn,Q,Xd),heap_Time_return(A,Xd),Q) ).

% return_wp_rule
tff(fact_4285_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_dm(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule
tff(fact_4286_cons__pre__rule,axiom,
    ! [A: $tType,P: assn,P2: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( entails(P,P2)
     => ( hoare_hoare_triple(A,P2,C2,Q)
       => hoare_hoare_triple(A,P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_4287_norm__pre__pure__rule1,axiom,
    ! [A: $tType,B2: $o,P: assn,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B2)
       => hoare_hoare_triple(A,P,F2,Q) )
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F2,Q) ) ).

% norm_pre_pure_rule1
tff(fact_4288_norm__pre__pure__rule2,axiom,
    ! [A: $tType,B2: $o,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B2)
       => hoare_hoare_triple(A,one_one(assn),F2,Q) )
     => hoare_hoare_triple(A,pure_assn((B2)),F2,Q) ) ).

% norm_pre_pure_rule2
tff(fact_4289_return__sp__rule,axiom,
    ! [A: $tType,P: assn,Xd: A] : hoare_hoare_triple(A,P,heap_Time_return(A,Xd),aa(A,fun(A,assn),aTP_Lamp_dq(assn,fun(A,fun(A,assn)),P),Xd)) ).

% return_sp_rule
tff(fact_4290_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(divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
            $ite(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_4291_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),array_nth(A,A2,I),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_dr(nat,fun(list(A),fun(array(A),fun(A,assn))),I),Xs),A2)) ) ) ).

% nth_rule
tff(fact_4292_sin__tan,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( sin(real,Xd) = divide_divide(real,aa(real,real,tan(real),Xd),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_4293_cos__tan,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( cos(real,Xd) = divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_4294_real__sqrt__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa(real,real,sqrt,Xd) = aa(real,real,sqrt,Y) )
    <=> ( Xd = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_4295_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xd: real] :
      ( ( aa(real,real,sqrt,Xd) = zero_zero(real) )
    <=> ( Xd = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_4296_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_4297_real__sqrt__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,Xd)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less(real,Xd),Y) ) ).

% real_sqrt_less_iff
tff(fact_4298_real__sqrt__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xd)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ).

% real_sqrt_le_iff
tff(fact_4299_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_4300_real__sqrt__eq__1__iff,axiom,
    ! [Xd: real] :
      ( ( aa(real,real,sqrt,Xd) = one_one(real) )
    <=> ( Xd = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_4301_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_4302_real__sqrt__lt__0__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xd),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_4303_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_4304_real__sqrt__le__0__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xd),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_4305_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_4306_real__sqrt__lt__1__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,Xd)),one_one(real))
    <=> aa(real,$o,ord_less(real,Xd),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_4307_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less_eq(real,one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_4308_real__sqrt__le__1__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xd)),one_one(real))
    <=> aa(real,$o,ord_less_eq(real,Xd),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_4309_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)) = abs_abs(real,A2) ).

% real_sqrt_mult_self
tff(fact_4310_real__sqrt__abs2,axiom,
    ! [Xd: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Xd)) = abs_abs(real,Xd) ).

% real_sqrt_abs2
tff(fact_4311_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_4312_real__sqrt__abs,axiom,
    ! [Xd: real] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = abs_abs(real,Xd) ).

% real_sqrt_abs
tff(fact_4313_real__sqrt__pow2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xd ) ) ).

% real_sqrt_pow2
tff(fact_4314_real__sqrt__pow2__iff,axiom,
    ! [Xd: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xd )
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd) ) ).

% real_sqrt_pow2_iff
tff(fact_4315_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xd: 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),Xd),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),Xd),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_4316_real__sqrt__le__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,Xd),Y)
     => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xd)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_4317_real__sqrt__mult,axiom,
    ! [Xd: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,Xd)),aa(real,real,sqrt,Y)) ).

% real_sqrt_mult
tff(fact_4318_real__sqrt__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xd),Y)
     => aa(real,$o,ord_less(real,aa(real,real,sqrt,Xd)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_4319_real__sqrt__divide,axiom,
    ! [Xd: real,Y: real] : aa(real,real,sqrt,divide_divide(real,Xd,Y)) = divide_divide(real,aa(real,real,sqrt,Xd),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_4320_real__sqrt__power,axiom,
    ! [Xd: real,K: nat] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xd)),K) ).

% real_sqrt_power
tff(fact_4321_divmod__integer_H__def,axiom,
    ! [M: num,Nb: num] : unique8689654367752047608divmod(code_integer,M,Nb) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(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))),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_4322_real__sqrt__minus,axiom,
    ! [Xd: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),Xd)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,Xd)) ).

% real_sqrt_minus
tff(fact_4323_Code__Numeral_Oset__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xd: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),Xaa),code_integer_of_int(Xd)) = code_integer_of_int(aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xaa),Xd)) ).

% Code_Numeral.set_bit_integer.abs_eq
tff(fact_4324_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: int] : divide_divide(code_integer,code_integer_of_int(Xaa),code_integer_of_int(Xd)) = code_integer_of_int(divide_divide(int,Xaa,Xd)) ).

% divide_integer.abs_eq
tff(fact_4325_modulo__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: int] : modulo_modulo(code_integer,code_integer_of_int(Xaa),code_integer_of_int(Xd)) = code_integer_of_int(modulo_modulo(int,Xaa,Xd)) ).

% modulo_integer.abs_eq
tff(fact_4326_real__sqrt__gt__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,sqrt,Xd)) ) ).

% real_sqrt_gt_zero
tff(fact_4327_real__sqrt__ge__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,Xd)) ) ).

% real_sqrt_ge_zero
tff(fact_4328_real__sqrt__eq__zero__cancel,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( ( aa(real,real,sqrt,Xd) = zero_zero(real) )
       => ( Xd = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_4329_real__sqrt__ge__one,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
     => aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,sqrt,Xd)) ) ).

% real_sqrt_ge_one
tff(fact_4330_zero__integer__def,axiom,
    zero_zero(code_integer) = code_integer_of_int(zero_zero(int)) ).

% zero_integer_def
tff(fact_4331_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: int] :
      ( aa(code_integer,$o,ord_less(code_integer,code_integer_of_int(Xaa)),code_integer_of_int(Xd))
    <=> aa(int,$o,ord_less(int,Xaa),Xd) ) ).

% less_integer.abs_eq
tff(fact_4332_one__integer__def,axiom,
    one_one(code_integer) = code_integer_of_int(one_one(int)) ).

% one_integer_def
tff(fact_4333_real__div__sqrt,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( divide_divide(real,Xd,aa(real,real,sqrt,Xd)) = aa(real,real,sqrt,Xd) ) ) ).

% real_div_sqrt
tff(fact_4334_sqrt__add__le__add__sqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,Xd)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_4335_le__real__sqrt__sumsq,axiom,
    ! [Xd: real,Y: real] : aa(real,$o,ord_less_eq(real,Xd),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),Xd),Xd)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_4336_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_4337_real__less__rsqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,ord_less(real,Xd),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_4338_sqrt__le__D,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xd)),Y)
     => aa(real,$o,ord_less_eq(real,Xd),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_4339_real__le__rsqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,ord_less_eq(real,Xd),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_4340_real__le__lsqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,Xd),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xd)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_4341_real__sqrt__unique,axiom,
    ! [Y: real,Xd: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xd )
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,real,sqrt,Xd) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_4342_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),U)
     => aa(real,$o,ord_less(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_4343_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xd: 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),Xd),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))))) = Xd )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_4344_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xd: 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),Xd),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 )
     => ( Xd = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_4345_real__sqrt__sum__squares__ge1,axiom,
    ! [Xd: real,Y: real] : aa(real,$o,ord_less_eq(real,Xd),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),Xd),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_4346_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xd: real] : aa(real,$o,ord_less_eq(real,Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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_4347_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D: real] : aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),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),D)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),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),D),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_4348_sqrt__ge__absD,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),aa(real,real,sqrt,Y))
     => aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_4349_cos__45,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = 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_4350_sin__45,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = 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_4351_tan__60,axiom,
    aa(real,real,tan(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_4352_real__less__lsqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,Xd),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,ord_less(real,aa(real,real,sqrt,Xd)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_4353_sqrt__sum__squares__le__sum,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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),Xd),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_4354_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))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_4355_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xd: real,Y: real] : aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),abs_abs(real,Xd)),abs_abs(real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_4356_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xd: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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_4357_real__sqrt__ge__abs1,axiom,
    ! [Xd: real,Y: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),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),Xd),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_4358_ln__sqrt,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xd)) = divide_divide(real,aa(real,real,ln_ln(real),Xd),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_4359_cos__30,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = 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_4360_sin__60,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = 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_4361_arsinh__real__def,axiom,
    ! [Xd: real] : aa(real,real,arsinh(real),Xd) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_4362_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xd)),Nb) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_4363_arsinh__real__aux,axiom,
    ! [Xd: real] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_4364_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xd: real,Y: real,Xaa: real,Ya: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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_4365_arith__geo__mean__sqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Y))),divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_4366_powr__half__sqrt,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,Xd) ) ) ).

% powr_half_sqrt
tff(fact_4367_tan__30,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = 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_4368_cos__x__y__le__one,axiom,
    ! [Xd: real,Y: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,divide_divide(real,Xd,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),Xd),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_4369_real__sqrt__sum__squares__less,axiom,
    ! [Xd: real,U: real,Y: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,ord_less(real,abs_abs(real,Y)),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),Xd),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_4370_arcosh__real__def,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
     => ( aa(real,real,arcosh(real),Xd) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_4371_cos__arctan,axiom,
    ! [Xd: real] : cos(real,aa(real,real,arctan,Xd)) = 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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% cos_arctan
tff(fact_4372_sin__arctan,axiom,
    ! [Xd: real] : sin(real,aa(real,real,arctan,Xd)) = divide_divide(real,Xd,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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% sin_arctan
tff(fact_4373_sqrt__sum__squares__half__less,axiom,
    ! [Xd: real,U: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xd),divide_divide(real,U,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,ord_less(real,Y),divide_divide(real,U,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
         => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
           => aa(real,$o,ord_less(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),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_4374_sin__cos__sqrt,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,Xd))
     => ( sin(real,Xd) = 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,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_4375_arctan__half,axiom,
    ! [Xd: real] : aa(real,real,arctan,Xd) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,divide_divide(real,Xd,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),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_4376_sin__paired,axiom,
    ! [Xd: real] : sums(real,aTP_Lamp_ds(real,fun(nat,real),Xd),sin(real,Xd)) ).

% sin_paired
tff(fact_4377_and__int_Opelims,axiom,
    ! [Xd: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xd),Xaa) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xd),Xaa))
       => ~ ( ( Y = $ite(
                  ( member(int,Xd,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))
                  & member(int,Xaa,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xd)
                      & ~ aa(int,$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))),Xd)
                        & ~ 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),divide_divide(int,Xd,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xaa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xd),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_4378_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(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) = $ite(
            ( member(int,K,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))
            & member(int,L,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_4379_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_dt(array(A),fun(list(A),fun(list(A),assn)),A2),Xs)) ) ).

% freeze_rule
tff(fact_4380_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_cp(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_4381_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xd: A] :
          ( sums(A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),A2),Xd)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xd ) ) ) ).

% powser_sums_zero_iff
tff(fact_4382_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( sums(A,F2,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_cs(fun(nat,A),fun(A,fun(nat,A)),F2),C2),divide_divide(A,A2,C2)) ) ) ).

% sums_divide
tff(fact_4383_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( sums(A,F2,A2)
         => ( sums(A,G,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_4384_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_co(nat,fun(fun(nat,A),fun(nat,A)),I),F2),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_4385_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(nat,A,F2,N) = zero_zero(A)
         => sums(A,F2,zero_zero(A)) ) ) ).

% sums_0
tff(fact_4386_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( sums(A,F2,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ).

% sums_mult2
tff(fact_4387_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( sums(A,F2,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_cy(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ).

% sums_mult
tff(fact_4388_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_du(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),D),C2))
          <=> sums(A,F2,D) ) ) ) ).

% sums_mult2_iff
tff(fact_4389_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dv(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D))
          <=> sums(A,F2,D) ) ) ) ).

% sums_mult_iff
tff(fact_4390_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cr(A,fun(fun(nat,A),fun(nat,A)),C2),F2),A2)
         => ( ( C2 != zero_zero(A) )
           => sums(A,F2,divide_divide(A,A2,C2)) ) ) ) ).

% sums_mult_D
tff(fact_4391_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),F2),S)
           => sums(A,F2,S) ) ) ) ).

% sums_Suc_imp
tff(fact_4392_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),F2),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_4393_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_dw(fun(nat,A),fun(nat,A),F2),L)
         => sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_4394_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F2: fun(nat,A),S: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,I3),Nb)
             => ( aa(nat,A,F2,I3) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dx(nat,fun(fun(nat,A),fun(nat,A)),Nb),F2),S)
          <=> sums(A,F2,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_4395_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_dy(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_4396_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_4397_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))
         => sums(A,aa(A,fun(nat,A),power_power(A),C2),divide_divide(A,one_one(A),aa(A,A,minus_minus(A,one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_4398_power__half__series,axiom,
    sums(real,aTP_Lamp_dz(nat,real),one_one(real)) ).

% power_half_series
tff(fact_4399_sums__if_H,axiom,
    ! [G: fun(nat,real),Xd: real] :
      ( sums(real,G,Xd)
     => sums(real,aTP_Lamp_ea(fun(nat,real),fun(nat,real),G),Xd) ) ).

% sums_if'
tff(fact_4400_sums__if,axiom,
    ! [G: fun(nat,real),Xd: real,F2: fun(nat,real),Y: real] :
      ( sums(real,G,Xd)
     => ( sums(real,F2,Y)
       => sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_eb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y)) ) ) ).

% sums_if
tff(fact_4401_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [K2: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K2),L3))
           => ( ( ~ ( member(int,K2,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))
                    & member(int,L3,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_4402_cos__paired,axiom,
    ! [Xd: real] : sums(real,aTP_Lamp_ec(real,fun(nat,real),Xd),cos(real,Xd)) ).

% cos_paired
tff(fact_4403_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))
         => sums(A,aTP_Lamp_ed(A,fun(nat,A),Z),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_4404_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [I3: int,J3: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I3),J3))
           => ( ( aa(int,$o,ord_less_eq(int,I3),J3)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J3) )
             => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_4405_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xd: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ee(fun(nat,A),fun(A,fun(nat,A)),C2),Xd))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ef(fun(nat,A),fun(A,fun(nat,A)),C2),Xd),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ee(fun(nat,A),fun(A,fun(nat,A)),C2),Xd))) ) ) ).

% diffs_equiv
tff(fact_4406_cos__arcsin,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xd)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_4407_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xd: A,M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(
            aa(nat,$o,ord_less(nat,Nb),M),
            zero_zero(A),
            $ite(Xd = 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)),divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(nat,nat,suc,Nb))),aa(A,A,minus_minus(A,one_one(A)),Xd))) ) ) ).

% sum_gp
tff(fact_4408_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_4409_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_eg(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_4410_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_4411_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),F3)
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),F3) = zero_zero(B) )
          <=> ! [X5: A] :
                ( member(A,X5,F3)
               => ( aa(A,B,F2,X5) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4412_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_4413_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_eh(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_4414_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ei(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_4415_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xd: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xd,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),insert(A,Xd,A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xd)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_4416_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ej(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_4417_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_4418_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_4419_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_4420_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_ek(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
            ( aa(set(nat),$o,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_4421_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arcsin_1
tff(fact_4422_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_el(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D)),A3) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A3)
            & member(nat,zero_zero(nat),A3) ),
            divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_4423_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arcsin_minus_1
tff(fact_4424_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A3: set(B),R2: A] : divide_divide(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_em(fun(B,A),fun(A,fun(B,A)),F2),R2)),A3) ) ).

% sum_divide_distrib
tff(fact_4425_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_bc(fun(B,A),fun(A,fun(B,A)),F2),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3),A2) ) ).

% mod_sum_eq
tff(fact_4426_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,F2: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_en(A,fun(fun(B,A),fun(B,A)),R2),F2)),A3) ) ).

% sum_distrib_left
tff(fact_4427_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_eo(fun(B,A),fun(A,fun(B,A)),F2),R2)),A3) ) ).

% sum_distrib_right
tff(fact_4428_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),B4)) = 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_eq(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B4)),A3) ) ).

% sum_product
tff(fact_4429_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_er(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_4430_dvd__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),D: B,F2: fun(A,B)] :
          ( ! [A5: A] :
              ( member(A,A5,A3)
             => aa(B,$o,dvd_dvd(B,D),aa(A,B,F2,A5)) )
         => aa(B,$o,dvd_dvd(B,D),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ).

% dvd_sum
tff(fact_4431_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) != zero_zero(A) )
         => ~ ! [A5: B] :
                ( member(B,A5,A3)
               => ( aa(B,A,G,A5) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4432_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => ( aa(A,B,G,X) = zero_zero(B) ) )
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_4433_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),zero_zero(B)) )
         => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_4434_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X)) )
         => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ).

% sum_nonneg
tff(fact_4435_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A3)
         => ( ! [X: nat] :
                ( member(nat,aa(nat,nat,suc,X),A3)
               => ( aa(nat,A,F2,aa(nat,nat,suc,X)) = aa(nat,A,G,aa(nat,nat,suc,X)) ) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),A3) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_4436_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_es(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_4437_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_et(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4438_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_et(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_4439_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_eu(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_4440_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_eu(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_4441_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3) = zero_zero(B) )
            <=> ! [X5: A] :
                  ( member(A,X5,A3)
                 => ( aa(A,B,F2,X5) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4442_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),T: set(B),G: fun(B,C),I: fun(B,A),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),T)
           => ( ! [X: B] :
                  ( member(B,X,T)
                 => aa(C,$o,ord_less_eq(C,zero_zero(C)),aa(B,C,G,X)) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ? [Xa: B] :
                        ( member(B,Xa,T)
                        & ( aa(B,A,I,Xa) = X )
                        & aa(C,$o,ord_less_eq(C,aa(A,C,F2,X)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,ord_less_eq(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,F2),S)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T)) ) ) ) ) ) ).

% sum_le_included
tff(fact_4443_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,G,X)) )
           => ( ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(B,$o,ord_less(B,aa(A,B,F2,X4)),aa(A,B,G,X4)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4444_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X22),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S3)
             => ( ! [X: B] :
                    ( member(B,X,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X)),aa(B,A,G,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S3)) ) ) ) ) ) ).

% sum.related
tff(fact_4445_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A] :
                  ( member(A,X,A3)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X)),aa(A,B,G,X)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono
tff(fact_4446_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C2: A,B2: A,D: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D )
           => ( ! [X: A] :
                  ( aa(A,$o,ord_less_eq(A,C2),X)
                 => ( aa(A,$o,ord_less(A,X),D)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( 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,D)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4447_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xd: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),insert(A,Xd,A3)) = $ite(member(A,Xd,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,Xd)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3))) ) ) ) ).

% sum.insert_if
tff(fact_4448_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T4: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( ! [A5: A] :
                  ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S3),S4))
                 => ( aa(B,A,I,aa(A,B,J,A5)) = A5 ) )
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S3),S4))
                   => member(B,aa(A,B,J,A5),aa(set(B),set(B),minus_minus(set(B),T4),T3)) )
               => ( ! [B3: B] :
                      ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T4),T3))
                     => ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T4),T3))
                       => member(A,aa(B,A,I,B3),aa(set(A),set(A),minus_minus(set(A),S3),S4)) )
                   => ( ! [A5: A] :
                          ( member(A,A5,S4)
                         => ( aa(A,C,G,A5) = zero_zero(C) ) )
                     => ( ! [B3: B] :
                            ( member(B,B3,T3)
                           => ( aa(B,C,H,B3) = zero_zero(C) ) )
                       => ( ! [A5: A] :
                              ( member(A,A5,S3)
                             => ( aa(B,C,H,aa(A,B,J,A5)) = aa(A,C,G,A5) ) )
                         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,G),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,H),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4449_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ( aa(nat,$o,ord_less_eq(nat,Nb),P3)
           => ( 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,P3))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4450_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),B4: B,I: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I3: A] :
                ( member(A,I3,S)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S) = B4 )
             => ( member(A,I,S)
               => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I)),B4) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4451_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I3: A] :
                ( member(A,I3,S)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S) = zero_zero(B) )
             => ( member(A,I,S)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4452_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,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_ev(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4453_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xd: A,M: nat,I6: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ew(A,fun(nat,fun(nat,A)),Xd),M)),I6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),I6)) ) ).

% sum_power_add
tff(fact_4454_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_ex(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or1337092689740270186AtMost(nat,Nb,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_4455_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A3)
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ct(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),A3)) ) ) ).

% sums_If_finite_set
tff(fact_4456_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),collect(nat,P))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cu(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),collect(nat,P))) ) ) ).

% sums_If_finite
tff(fact_4457_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N2)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => sums(A,F2,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N2)) ) ) ) ).

% sums_finite
tff(fact_4458_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N2)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N2) ) ) ) ) ).

% suminf_finite
tff(fact_4459_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( member(A,I,I6)
           => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I6)
                   => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I6)) ) ) ) ) ) ).

% sum_pos2
tff(fact_4460_arcsin__minus,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),Xd)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,Xd)) ) ) ) ).

% arcsin_minus
tff(fact_4461_arcsin__le__arcsin,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Xd)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_4462_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ( I6 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I6)
                 => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I6)) ) ) ) ) ).

% sum_pos
tff(fact_4463_arcsin__le__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Xd)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_4464_arcsin__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
       => ( ( aa(real,real,arcsin,Xd) = aa(real,real,arcsin,Y) )
        <=> ( Xd = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_4465_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C6: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C6)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),C6)
           => ( aa(set(A),$o,ord_less_eq(set(A),B4),C6)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C6),A3))
                   => ( aa(A,B,G,A5) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C6),B4))
                     => ( 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),B4) )
                  <=> ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C6) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C6) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4466_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C6: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C6)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),C6)
           => ( aa(set(A),$o,ord_less_eq(set(A),B4),C6)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C6),A3))
                   => ( aa(A,B,G,A5) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C6),B4))
                     => ( aa(A,B,H,B3) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C6) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C6) )
                   => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4467_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T4) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4468_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T4) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S3) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4469_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,H,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4470_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T4) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4471_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B4: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),B4),A3)
         => ( aa(set(A),$o,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),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_4472_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_4473_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_4474_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_4475_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_4476_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_4477_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$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_4478_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_4479_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_4480_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_4481_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : aa(nat,A,diffs(A,C2),X4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X4))),aa(nat,A,C2,aa(nat,nat,suc,X4))) ) ).

% diffs_def
tff(fact_4482_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_et(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_4483_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ey(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Nb)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4484_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ey(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F2,aa(nat,nat,suc,Nb))),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_4485_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_ez(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or7035219750837199246ssThan(nat,Nb,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_4486_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_fa(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_fc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% sum.nested_swap
tff(fact_4487_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S3: A,A3: set(nat),S4: A,F2: fun(nat,A)] :
          ( sums(A,G,S3)
         => ( aa(set(nat),$o,finite_finite2(nat),A3)
           => ( ( S4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fe(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_4488_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ff(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4489_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B4),A3))
                 => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,B3)) )
             => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4)) ) ) ) ) ).

% sum_mono2
tff(fact_4490_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xd: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xd,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,Xd)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_4491_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xd: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),insert(A,Xd,A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xd)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_4492_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xd: A] :
          ( ! [X: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Xd)) ) ) ).

% termdiff_converges_all
tff(fact_4493_arcsin__less__arcsin,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less(real,aa(real,real,arcsin,Xd)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_4494_arcsin__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,aa(real,real,arcsin,Xd)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,ord_less(real,Xd),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_4495_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$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),P3))) = 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),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_4496_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_4497_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I6: set(nat)] :
          ( summable(A,F2)
         => ( aa(set(nat),$o,finite_finite2(nat),I6)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I6))
                 => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N)) )
             => aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),I6)),suminf(A,F2)) ) ) ) ) ).

% sum_le_suminf
tff(fact_4498_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fi(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S3) = $ite(member(A,A2,S3),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),S3),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),S3),insert(A,A2,bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_4499_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_4500_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_ex(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_4501_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
           => ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),B4),A3))
             => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,B2))
               => ( ! [X: A] :
                      ( member(A,X,B4)
                     => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X)) )
                 => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4502_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A3: set(A),F2: fun(A,B)] :
          ( member(A,I,A3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,I,bot_bot(set(A)))))
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X)) )
           => ( aa(set(A),$o,finite_finite2(A),A3)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_4503_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),Xd: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Xd,I3)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xd),I6) = one_one(B) )
           => ( ! [I3: A] :
                  ( member(A,I3,I6)
                 => aa(B,$o,ord_less_eq(B,abs_abs(B,aa(B,B,minus_minus(B,aa(A,B,A2,I3)),B2))),Delta) )
             => aa(B,$o,ord_less_eq(B,abs_abs(B,aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_fj(fun(A,B),fun(fun(A,B),fun(A,B)),Xd),A2)),I6)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_4504_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [N7: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,N7),M4)
                 => ! [N6: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M4,N6)))),E3) ) ) ) ) ).

% summable_Cauchy
tff(fact_4505_cos__arcsin__nonzero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xd)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_4506_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fk(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(aa(nat,$o,ord_less_eq(nat,M),Nb),aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_4507_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fl(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Nb)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_4508_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( sums(A,F2,S)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S) ) ) ) ).

% sums_group
tff(fact_4509_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E: real] :
          ( summable(A,F2)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E)
           => ~ ! [N8: nat] :
                  ~ ! [M3: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,N8),M3)
                     => ! [N9: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,M3,N9)))),E) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_4510_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aTP_Lamp_as(nat,fun(nat,$o),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_4511_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Nb: nat,Xd: A] :
          ( aa(nat,$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)),Xd)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_or1337092689740270186AtMost(nat,M,Nb))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(nat,nat,suc,Nb))) ) ) ) ).

% sum_gp_multiplied
tff(fact_4512_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_fn(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.in_pairs
tff(fact_4513_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,K6: real,C2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),K6)
         => ( ! [X: A] :
                ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),K6)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),X)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fo(A,fun(fun(nat,A),fun(nat,A)),Xd),C2)) ) ) ) ).

% termdiff_converges
tff(fact_4514_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_as(nat,fun(nat,$o),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_4515_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_bg(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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_4516_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_4517_Sum__Ico__nat,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_bg(nat,nat)),set_or7035219750837199246ssThan(nat,M,Nb)) = 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_4518_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_4519_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D: 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_fp(A,fun(A,fun(nat,A)),A2),D)),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)),D))) ) ).

% double_arith_series
tff(fact_4520_arith__series__nat,axiom,
    ! [A2: nat,D: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fq(nat,fun(nat,fun(nat,nat)),A2),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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),D))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_4521_Sum__Icc__nat,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_bg(nat,nat)),set_or1337092689740270186AtMost(nat,M,Nb)) = 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_4522_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,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less(real,aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_4523_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_4524_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_4525_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_4526_arcsin__sin,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,sin(real,Xd)) = Xd ) ) ) ).

% arcsin_sin
tff(fact_4527_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_4528_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)) = 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_4529_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fr(A,fun(A,fun(nat,A)),A2),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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)),D))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_4530_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xd: A,M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) = $ite(Xd = 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)),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),Xd),M)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(nat,nat,suc,Nb)))),aa(A,A,minus_minus(A,one_one(A)),Xd))) ) ).

% sum_gp_offset
tff(fact_4531_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_4532_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_4533_arcsin__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Xd)),Y)
            <=> aa(real,$o,ord_less_eq(real,Xd),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_4534_le__arcsin__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,arcsin,Xd))
            <=> aa(real,$o,ord_less_eq(real,sin(real,Y)),Xd) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_4535_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)) = 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_4536_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I3),J3)
             => ( aa(nat,$o,ord_less(nat,J3),Nb)
               => aa(A,$o,ord_less_eq(A,aa(nat,A,A2,I3)),aa(nat,A,A2,J3)) ) )
         => ( ! [I3: nat,J3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,I3),J3)
               => ( aa(nat,$o,ord_less(nat,J3),Nb)
                 => aa(A,$o,ord_less_eq(A,aa(nat,A,B2,J3)),aa(nat,A,B2,I3)) ) )
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fs(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_4537_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J3: nat] :
          ( aa(nat,$o,ord_less_eq(nat,I3),J3)
         => ( aa(nat,$o,ord_less(nat,J3),Nb)
           => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,A2,I3)),aa(nat,nat,A2,J3)) ) )
     => ( ! [I3: nat,J3: nat] :
            ( aa(nat,$o,ord_less_eq(nat,I3),J3)
           => ( aa(nat,$o,ord_less(nat,J3),Nb)
             => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,B2,J3)),aa(nat,nat,B2,I3)) ) )
       => aa(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_ft(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_4538_Maclaurin__cos__expansion2,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ? [T6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T6)
            & aa(real,$o,ord_less(real,T6),Xd)
            & ( cos(real,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fu(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(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),Xd),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_4539_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,Xd),zero_zero(real))
       => ? [T6: real] :
            ( aa(real,$o,ord_less(real,Xd),T6)
            & aa(real,$o,ord_less(real,T6),zero_zero(real))
            & ( cos(real,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fu(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(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),Xd),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_4540_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_4541_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_4542_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_4543_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_4544_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fv(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_4545_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_fw(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_4546_sum__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),multiset(B),groups7311177749621191930dd_sum(A,multiset(B),F2),A3) = zero_zero(multiset(B)) )
      <=> ! [X5: A] :
            ( member(A,X5,A3)
           => ( aa(A,multiset(B),F2,X5) = zero_zero(multiset(B)) ) ) ) ) ).

% sum_eq_empty_iff
tff(fact_4547_lessThan__atLeast0,axiom,
    ! [Nb: nat] : set_ord_lessThan(nat,Nb) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ).

% lessThan_atLeast0
tff(fact_4548_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_4549_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( set_ord_lessThan(nat,Nb) = bot_bot(set(nat)) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_4550_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = insert(nat,K,set_ord_lessThan(nat,K)) ).

% lessThan_Suc
tff(fact_4551_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F2: 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,F2),A3)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_fx(A,fun(fun(B,nat),fun(B,A)),C2),F2),A3) ) ).

% power_sum
tff(fact_4552_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),Nb: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = aa(nat,nat,suc,Nb) )
     => ? [X: A] :
          ( member(A,X,A3)
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,F2,X)) ) ) ).

% sum_SucD
tff(fact_4553_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X5: A] :
            ( member(A,X5,A3)
            & ( aa(A,nat,F2,X5) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X5 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4554_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = one_one(nat) )
      <=> ? [X5: A] :
            ( member(A,X5,A3)
            & ( aa(A,nat,F2,X5) = one_one(nat) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X5 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4555_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K)) = insert(nat,pred_numeral(K),set_ord_lessThan(nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_4556_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_fy(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_4557_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_fz(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_4558_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Nb)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_ga(complex,complex)),collect(complex,aTP_Lamp_gb(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_4559_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Nb)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_ga(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_ab(nat,fun(complex,fun(complex,$o)),Nb),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_4560_ln__prod,axiom,
    ! [A: $tType,I6: set(A),F2: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I6)
     => ( ! [I3: A] :
            ( member(A,I3,I6)
           => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,I3)) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,F2,I6)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_gc(fun(A,real),fun(A,real),F2)),I6) ) ) ) ).

% ln_prod
tff(fact_4561_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_et(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_4562_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ey(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_4563_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gd(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_4564_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Nb: nat,R2: A] : aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),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)),R2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ge(fun(nat,A),fun(A,fun(nat,A)),F2),R2)),set_ord_lessThan(nat,Nb)) ) ).

% sumr_diff_mult_const2
tff(fact_4565_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_bd(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4566_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xd: A] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),Xd)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_4567_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_et(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_4568_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_et(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_4569_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_iff_shift
tff(fact_4570_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb))))
        <=> sums(A,F2,S) ) ) ).

% sums_iff_shift'
tff(fact_4571_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A,Nb: nat] :
          ( sums(A,F2,S)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_4572_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_bd(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Nb)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4573_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_gf(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_4574_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_gg(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_4575_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I6: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Z,I3))),one_one(real)) )
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,W,I3))),one_one(real)) )
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,groups7121269368397514597t_prod(A,B,Z,I6)),groups7121269368397514597t_prod(A,B,W,I6)))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_gh(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I6)) ) ) ) ).

% norm_prod_diff
tff(fact_4576_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xd)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_ord_lessThan(nat,Nb))) ) ).

% one_diff_power_eq
tff(fact_4577_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xd),one_one(A))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_ord_lessThan(nat,Nb))) ) ).

% power_diff_1_eq
tff(fact_4578_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xd: A,Nb: nat] :
          ( ( Xd != one_one(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_ord_lessThan(nat,Nb)) = divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)),one_one(A)),aa(A,A,minus_minus(A,Xd),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_4579_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,F2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_4580_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_4581_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat] :
          ( summable(A,F2)
         => ( ! [M2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nb),M2)
               => aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,M2)) )
           => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_4582_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_gi(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_gj(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_4583_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_ord_lessThan(nat,Nb)) = $ite(Xd = one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb),divide_divide(A,aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)),aa(A,A,minus_minus(A,one_one(A)),Xd))) ) ).

% sum_gp_strict
tff(fact_4584_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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,Xd),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gk(A,fun(nat,fun(A,fun(nat,A))),Xd),Nb),Y)),set_ord_lessThan(nat,Nb))) ) ).

% power_diff_sumr2
tff(fact_4585_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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,Xd),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gl(A,fun(nat,fun(A,fun(nat,A))),Xd),Nb),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb)))) ) ).

% diff_power_eq_sum
tff(fact_4586_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)),insert(nat,zero_zero(nat),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_4587_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F2: fun(nat,A),K6: A,K: nat] :
          ( ! [P5: nat] :
              ( aa(nat,$o,ord_less(nat,P5),Nb)
             => aa(A,$o,ord_less_eq(A,aa(nat,A,F2,P5)),K6) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),K6)
           => aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),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_4588_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nb),M2)
               => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,M2)) )
           => ( aa(nat,$o,ord_less_eq(nat,Nb),I)
             => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,I))
               => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Nb))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_4589_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xd)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gm(A,fun(nat,fun(nat,A)),Xd),Nb)),set_ord_lessThan(nat,Nb))) ) ).

% one_diff_power_eq'
tff(fact_4590_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xd: real,Nb: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xd = 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_gn(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xd),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_4591_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ? [B9: real] : aa(real,real,F2,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_go(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),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_4592_sum__split__even__odd,axiom,
    ! [F2: 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_gp(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),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_gq(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,Nb))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gr(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,Nb))) ).

% sum_split_even_odd
tff(fact_4593_Maclaurin__exp__le,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T6: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,T6)),abs_abs(real,Xd))
      & ( aa(real,real,exp(real),Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gs(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,exp(real),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_4594_Sum__Icc__int,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,ord_less_eq(int,M),Nb)
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_bp(int,int)),set_or1337092689740270186AtMost(int,M,Nb)) = 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_4595_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D2: nat] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)))),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)),one_one(nat))))))
       => aa(real,$o,ord_less(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F2),set_ord_lessThan(nat,K))),suminf(real,F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_4596_Maclaurin__exp__lt,axiom,
    ! [Xd: real,Nb: nat] :
      ( ( Xd != zero_zero(real) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ? [T6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),abs_abs(real,T6))
            & aa(real,$o,ord_less(real,abs_abs(real,T6)),abs_abs(real,Xd))
            & ( aa(real,real,exp(real),Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gs(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,exp(real),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_4597_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(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_gu(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_4598_Maclaurin__sin__expansion,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T6: real] : sin(real,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gv(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(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),Xd),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_4599_Maclaurin__sin__expansion2,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T6: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,T6)),abs_abs(real,Xd))
      & ( sin(real,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gv(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(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),Xd),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_4600_Maclaurin__cos__expansion,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T6: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,T6)),abs_abs(real,Xd))
      & ( cos(real,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fu(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(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),Xd),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_4601_Maclaurin__sin__expansion4,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ? [T6: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),T6)
          & aa(real,$o,ord_less_eq(real,T6),Xd)
          & ( sin(real,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gv(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(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),Xd),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_4602_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ? [T6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T6)
            & aa(real,$o,ord_less(real,T6),Xd)
            & ( sin(real,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gv(real,fun(nat,real),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(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),Xd),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_4603_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_gw(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_4604_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_gx(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_4605_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_4606_sin__arccos,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xd)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_4607_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_4608_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_4609_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_4610_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_4611_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = insert(nat,zero_zero(nat),bot_bot(set(nat))) ).

% atMost_0
tff(fact_4612_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_4613_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arccos_0
tff(fact_4614_atMost__atLeast0,axiom,
    ! [Nb: nat] : set_ord_atMost(nat,Nb) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb) ).

% atMost_atLeast0
tff(fact_4615_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_4616_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = insert(nat,aa(nat,nat,suc,K),set_ord_atMost(nat,K)) ).

% atMost_Suc
tff(fact_4617_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K)) = insert(nat,aa(num,nat,numeral_numeral(nat),K),set_ord_atMost(nat,pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_4618_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$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_4619_arccos__le__arccos,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),aa(real,real,arccos,Xd)) ) ) ) ).

% arccos_le_arccos
tff(fact_4620_sum__choose__upper,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gy(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_4621_arccos__le__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Xd)),aa(real,real,arccos,Y))
        <=> aa(real,$o,ord_less_eq(real,Y),Xd) ) ) ) ).

% arccos_le_mono
tff(fact_4622_arccos__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
        & aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real)) )
     => ( ( aa(real,real,arccos,Xd) = aa(real,real,arccos,Y) )
      <=> ( Xd = Y ) ) ) ).

% arccos_eq_iff
tff(fact_4623_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_et(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_4624_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),I: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gd(fun(nat,A),fun(nat,A),F2)),set_ord_atMost(nat,I)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_4625_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,D: fun(nat,A)] :
          ( ! [X5: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gz(fun(nat,A),fun(A,fun(nat,A)),C2),X5)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gz(fun(nat,A),fun(A,fun(nat,A)),D),X5)),set_ord_atMost(nat,Nb))
        <=> ! [I2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I2),Nb)
             => ( aa(nat,A,C2,I2) = aa(nat,A,D,I2) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_4626_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_bd(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_4627_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B4: A] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,A2,N))
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_ord_atMost(nat,N))),B4)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_4628_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_ha(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_fc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_ord_lessThan(nat,Nb)) ) ).

% sum.nested_swap'
tff(fact_4629_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_hb(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_cm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb),set_ord_lessThan(nat,Nb)) ) ).

% prod.nested_swap'
tff(fact_4630_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hc(nat,fun(nat,nat),R2)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_4631_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hd(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_4632_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hd(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_4633_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_4634_arccos__less__arccos,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less(real,aa(real,real,arccos,Y)),aa(real,real,arccos,Xd)) ) ) ) ).

% arccos_less_arccos
tff(fact_4635_arccos__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,aa(real,real,arccos,Xd)),aa(real,real,arccos,Y))
        <=> aa(real,$o,ord_less(real,Y),Xd) ) ) ) ).

% arccos_less_mono
tff(fact_4636_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_4637_arccos__cos,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),pi)
       => ( aa(real,real,arccos,cos(real,Xd)) = Xd ) ) ) ).

% arccos_cos
tff(fact_4638_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
     => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_4639_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( ! [X5: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gz(fun(nat,A),fun(A,fun(nat,A)),C2),X5)),set_ord_atMost(nat,Nb)) = zero_zero(A)
        <=> ! [I2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I2),Nb)
             => ( aa(nat,A,C2,I2) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_4640_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_he(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,Nb)) = zero_zero(A)
         => ( aa(nat,$o,ord_less_eq(nat,K),Nb)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_4641_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_et(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% sum.atMost_shift
tff(fact_4642_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),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,F2),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),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_4643_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_bd(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Nb))) ) ).

% prod.atMost_shift
tff(fact_4644_sum__choose__diagonal,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Nb)
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hf(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_4645_vandermonde,axiom,
    ! [M: nat,Nb: nat,R2: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hg(nat,fun(nat,fun(nat,fun(nat,nat))),M),Nb),R2)),set_ord_atMost(nat,R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),R2) ).

% vandermonde
tff(fact_4646_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less(real,Y),one_one(real))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less(real,aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_4647_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_4648_sin__arccos__nonzero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xd)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_4649_arccos__cos2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Xd)
       => ( aa(real,real,arccos,cos(real,Xd)) = aa(real,real,uminus_uminus(real),Xd) ) ) ) ).

% arccos_cos2
tff(fact_4650_arccos__minus,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xd)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,Xd)) ) ) ) ).

% arccos_minus
tff(fact_4651_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xd)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),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),Xd),aa(nat,nat,suc,Nb))) ) ).

% sum_gp_basic
tff(fact_4652_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_hh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
        <=> ? [I2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I2),Nb)
              & ( aa(nat,A,C2,I2) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_4653_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,ord_less_eq(nat,K),Nb)
           => aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_hh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4654_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_hi(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),B3),Z3)),set_ord_lessThan(nat,Nb))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4655_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Nb: nat,A2: A] :
        ? [B3: fun(nat,A)] :
        ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),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,Z3),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),B3),Z3)),set_ord_lessThan(nat,Nb)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Nb))) ) ).

% polyfun_linear_factor
tff(fact_4656_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Nb: nat,Xd: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),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),Xd),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,Nb),M)))) ) ) ) ).

% sum_power_shift
tff(fact_4657_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)),insert(nat,zero_zero(nat),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_4658_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_4659_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_hj(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_hj(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4660_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_hj(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_hj(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_hl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_4661_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_hm(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ).

% binomial
tff(fact_4662_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_fn(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% sum.in_pairs_0
tff(fact_4663_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xd: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,M),I3)
             => ( aa(nat,A,A2,I3) = zero_zero(A) ) )
         => ( ! [J3: nat] :
                ( aa(nat,$o,ord_less(nat,Nb),J3)
               => ( aa(nat,A,B2,J3) = 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_hi(fun(nat,A),fun(A,fun(nat,A)),A2),Xd)),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),B2),Xd)),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_ho(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xd)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_4664_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_bk(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Nb)) ) ).

% prod.in_pairs_0
tff(fact_4665_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: A] :
          ( ! [X5: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gz(fun(nat,A),fun(A,fun(nat,A)),C2),X5)),set_ord_atMost(nat,Nb)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X5: nat] :
                ( member(nat,X5,set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C2,X5) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4666_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi)
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_4667_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_hp(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% binomial_ring
tff(fact_4668_arccos__minus__abs,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xd)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,Xd)) ) ) ).

% arccos_minus_abs
tff(fact_4669_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_hq(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% pochhammer_binomial_sum
tff(fact_4670_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xd: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,M),I3)
         => ( aa(nat,nat,A2,I3) = zero_zero(nat) ) )
     => ( ! [J3: nat] :
            ( aa(nat,$o,ord_less(nat,Nb),J3)
           => ( aa(nat,nat,B2,J3) = 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_hr(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xd)),set_ord_atMost(nat,M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hr(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xd)),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_ht(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xd)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_4671_choose__square__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hu(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_4672_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_hj(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_hj(fun(nat,A),fun(nat,real),B2))
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_hl(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_4673_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),P3)
         => ( aa(nat,$o,ord_less_eq(nat,K),P3)
           => ( 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_hv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P3)) = 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_hw(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,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_4674_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),P3)
         => ( aa(nat,$o,ord_less_eq(nat,K),P3)
           => ( groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hx(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,P3)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hy(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,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4675_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,Z: A,A2: A] :
          ( aa(nat,$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_hz(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4676_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xd: A,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xd)),set_ord_atMost(nat,Nb)) = $ite(Xd = 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))),divide_divide(A,aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),aa(nat,nat,suc,Nb))),aa(A,A,minus_minus(A,one_one(A)),Xd))) ) ).

% sum_gp0
tff(fact_4677_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_ia(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4678_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xd: A,Y: A] :
          ( aa(nat,$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_hi(fun(nat,A),fun(A,fun(nat,A)),A2),Xd)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(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,Xd),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_ic(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xd),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4679_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_4680_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_4681_choose__linear__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_id(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_4682_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_ie(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4683_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E: real,C2: fun(nat,A),Nb: nat] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),E)
         => ? [M8: real] :
            ! [Z3: A] :
              ( aa(real,$o,ord_less_eq(real,M8),real_V7770717601297561774m_norm(A,Z3))
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_da(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,Nb)))),aa(real,real,aa(real,fun(real,real),times_times(real),E),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4684_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xd: A,Y: A] :
          ( aa(nat,$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_hi(fun(nat,A),fun(A,fun(nat,A)),A2),Xd)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(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,Xd),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_ig(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xd),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff
tff(fact_4685_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,cos(real,Theta)) != abs_abs(real,aa(real,real,minus_minus(real,Theta),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_4686_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ii(A,fun(A,fun(nat,A)),Xd),Y),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xd)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_4687_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ik(A,fun(A,fun(nat,A)),Xd),Y),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_4688_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_im(A,fun(A,fun(nat,A)),Xd),Y),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xd)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_4689_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_4690_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xd: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xd) )
        <=> ( ( A2 = B2 )
            | ( Xd = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_4691_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real] : aa(A,A,real_V8093663219630862766scaleR(A,A2),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_4692_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A2: real,Xd: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y)) ) ).

% mult_scaleR_left
tff(fact_4693_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [Xd: A,A2: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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),Xd),Y)) ) ).

% mult_scaleR_right
tff(fact_4694_scaleR__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xd: A,Y: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
        <=> ( ( Xd = Y )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_cancel_left
tff(fact_4695_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A] : aa(A,A,real_V8093663219630862766scaleR(A,one_one(real)),Xd) = Xd ) ).

% scaleR_one
tff(fact_4696_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xd: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xd)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)),Xd) ) ).

% scaleR_scaleR
tff(fact_4697_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xd: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( Xd = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_4698_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),Xd) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_4699_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_4700_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xd: real,Y: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,real_V8093663219630862766scaleR(A,Xd),Y)),Nb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) ) ).

% scaleR_power
tff(fact_4701_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real))),Xd) = aa(A,A,uminus_uminus(A),Xd) ) ).

% scaleR_minus1_left
tff(fact_4702_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_4703_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: real,Xd: A] : real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)) = aa(real,real,aa(real,fun(real,real),times_times(real),abs_abs(real,A2)),real_V7770717601297561774m_norm(A,Xd)) ) ).

% norm_scaleR
tff(fact_4704_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_4705_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,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,divide_divide(real,aa(num,real,numeral_numeral(real),W),aa(num,real,numeral_numeral(real),V))),A2) ) ).

% inverse_scaleR_times
tff(fact_4706_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,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,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_4707_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,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_4708_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A,A2: real,B2: real] :
          ( ( Xd != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xd) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_4709_scaleR__left__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xd: A,Y: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
           => ( Xd = Y ) ) ) ) ).

% scaleR_left_imp_eq
tff(fact_4710_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xd: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_distrib
tff(fact_4711_real__scaleR__def,axiom,
    ! [A2: real,Xd: real] : aa(real,real,real_V8093663219630862766scaleR(real,A2),Xd) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),Xd) ).

% real_scaleR_def
tff(fact_4712_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xd: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xd) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xd)) ) ).

% scaleR_left_distrib
tff(fact_4713_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),Y)),Xaa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Xd),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xaa)) ) ).

% scaleR_left.add
tff(fact_4714_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real] : real_Vector_of_real(A,R2) = aa(A,A,real_V8093663219630862766scaleR(A,R2),one_one(A)) ) ).

% of_real_def
tff(fact_4715_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real,Xd: A] : aa(A,A,real_V8093663219630862766scaleR(A,R2),Xd) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,R2)),Xd) ) ).

% scaleR_conv_of_real
tff(fact_4716_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xd: A] :
          ( aa(real,$o,ord_less_eq(real,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xd)) ) ) ) ).

% scaleR_right_mono
tff(fact_4717_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( aa(real,$o,ord_less_eq(real,B2),A2)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$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_4718_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,A2),B2) )
            & ( aa(real,$o,ord_less(real,C2),zero_zero(real))
             => aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4719_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,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4720_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,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,ord_less_eq(A,A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4721_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xd: A,Y: A,A2: real] :
          ( aa(A,$o,ord_less_eq(A,Xd),Y)
         => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_4722_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( aa(A,$o,ord_less_eq(A,B2),A2)
         => ( aa(real,$o,ord_less_eq(real,C2),zero_zero(real))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_4723_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V: real,A2: A,Xd: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V)),A2) = Xd )
        <=> $ite(V = zero_zero(real),Xd = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V),Xd)) ) ) ).

% vector_fraction_eq_iff
tff(fact_4724_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A,U: real,V: real,A2: A] :
          ( ( Xd = aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V)),A2) )
        <=> $ite(V = zero_zero(real),Xd = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V),Xd) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_4725_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),E)),C2)),D) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4726_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D))
        <=> aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,B2),A2)),E)),D)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4727_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_4728_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$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,ord_less_eq(A,B2),zero_zero(A)) )
            | ( aa(real,$o,ord_less(real,A2),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4729_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$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,ord_less_eq(A,zero_zero(A)),B2) )
            | ( aa(real,$o,ord_less(real,A2),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4730_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xd: A,Y: A] :
          ( aa(real,$o,ord_less_eq(real,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,Xd),Y)
           => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),B2)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
               => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4731_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D: A] :
          ( aa(real,$o,ord_less_eq(real,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4732_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xd: A] :
          ( ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
              & aa(A,$o,ord_less_eq(A,Xd),zero_zero(A)) )
            | ( aa(real,$o,ord_less_eq(real,A2),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd) ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_4733_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B2) )
            | ( aa(real,$o,ord_less_eq(real,A2),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,B2),zero_zero(A)) ) )
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_4734_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xd: A] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4735_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xd: A] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A2)
         => ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4736_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xd: A] :
          ( aa(real,$o,ord_less_eq(real,A2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4737_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(real,$o,ord_less_eq(real,A2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,B2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4738_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2))),Xd) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Xd) ) ).

% scaleR_2
tff(fact_4739_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xd: A,A2: real] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xd)
         => ( aa(real,$o,ord_less_eq(real,A2),one_one(real))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),Xd) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4740_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sums(A,aTP_Lamp_in(A,fun(nat,A),Xd),sin(A,Xd)) ) ).

% sin_converges
tff(fact_4741_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : sin(A,X4) = suminf(A,aTP_Lamp_in(A,fun(nat,A),X4)) ) ).

% sin_def
tff(fact_4742_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sums(A,aTP_Lamp_io(A,fun(nat,A),Xd),cos(A,Xd)) ) ).

% cos_converges
tff(fact_4743_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : cos(A,X4) = suminf(A,aTP_Lamp_io(A,fun(nat,A),X4)) ) ).

% cos_def
tff(fact_4744_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : summable(real,aTP_Lamp_ip(A,fun(nat,real),Xd)) ) ).

% summable_norm_sin
tff(fact_4745_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : summable(real,aTP_Lamp_iq(A,fun(nat,real),Xd)) ) ).

% summable_norm_cos
tff(fact_4746_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sums(A,aTP_Lamp_ir(A,fun(nat,A),Xd),sin(A,Xd)) ) ).

% sin_minus_converges
tff(fact_4747_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sums(A,aTP_Lamp_is(A,fun(nat,A),Xd),cos(A,Xd)) ) ).

% cos_minus_converges
tff(fact_4748_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_it(A,fun(nat,A),A2)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(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_4749_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_4750_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_iu(array(A),fun(list(A),fun(nat,assn)),A2),Xs)) ) ).

% length_rule
tff(fact_4751_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_4752_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_4753_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_4754_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_4755_time__array__len,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),H: heap_ext(product_unit)] : time_time(nat,array_len(A,P3),H) = one_one(nat) ) ).

% time_array_len
tff(fact_4756_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_4757_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_4758_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_4759_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_4760_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_4761_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),K)),A2)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(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_4762_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_4763_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_4764_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( aa(nat,$o,ord_less_eq(nat,K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,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_4765_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_iv(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_4766_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)),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_4767_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),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_4768_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_4769_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_4770_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_4771_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_4772_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = 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_4773_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = 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_4774_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_iw(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_4775_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_ix(A,fun(nat,fun(nat,A)),A2),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4776_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_iy(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_4777_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_iy(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_4778_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = divide_divide(A,groups7121269368397514597t_prod(nat,A,aTP_Lamp_iz(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4779_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_ja(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_4780_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,Xd: 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_jb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xd),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_jc(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xd),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4781_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)) = divide_divide(A,groups7121269368397514597t_prod(nat,A,aTP_Lamp_iz(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_4782_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),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_4783_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jd(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_4784_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,Xd: 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_jb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xd),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_je(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xd),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4785_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),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_jf(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_4786_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_it(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4787_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( topological_monoseq(A,X7)
        <=> ( ! [N6: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X7,N6)),aa(nat,A,X7,aa(nat,nat,suc,N6)))
            | ! [N6: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X7,aa(nat,nat,suc,N6))),aa(nat,A,X7,N6)) ) ) ) ).

% monoseq_Suc
tff(fact_4788_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X7,aa(nat,nat,suc,N))),aa(nat,A,X7,N))
         => topological_monoseq(A,X7) ) ) ).

% mono_SucI2
tff(fact_4789_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X7,N)),aa(nat,A,X7,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X7) ) ) ).

% mono_SucI1
tff(fact_4790_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : aa(A,A,exp(A),Xd) = 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)),Xd)),suminf(A,aTP_Lamp_jg(A,fun(nat,A),Xd))) ) ).

% exp_first_two_terms
tff(fact_4791_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_jh(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_4792_Maclaurin__sin__bound,axiom,
    ! [Xd: real,Nb: nat] : aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,sin(real,Xd)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gv(real,fun(nat,real),Xd)),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),abs_abs(real,Xd)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_4793_inverse__eq__iff__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% inverse_eq_iff_eq
tff(fact_4794_inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ).

% inverse_inverse_eq
tff(fact_4795_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_4796_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_4797_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_4798_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xd: A] :
          ( ( aa(A,A,inverse_inverse(A),Xd) = one_one(A) )
        <=> ( Xd = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_4799_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_4800_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),divide_divide(A,A2,B2)) = divide_divide(A,B2,A2) ) ).

% inverse_divide
tff(fact_4801_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ).

% inverse_minus_eq
tff(fact_4802_abs__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : abs_abs(A,aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),abs_abs(A,A2)) ) ).

% abs_inverse
tff(fact_4803_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) ) ).

% sgn_inverse
tff(fact_4804_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% inverse_sgn
tff(fact_4805_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4806_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4807_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_4808_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_4809_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_4810_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_4811_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,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4812_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,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4813_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_4814_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_4815_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)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_4816_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))) = 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_4817_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xd: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xd) = aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xd) = aa(A,A,aa(A,fun(A,A),times_times(A),Xd),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_4818_inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( A2 = B2 ) ) ) ).

% inverse_eq_imp_eq
tff(fact_4819_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_4820_real__sqrt__inverse,axiom,
    ! [Xd: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),Xd)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xd)) ).

% real_sqrt_inverse
tff(fact_4821_nonzero__of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xd: real] :
          ( ( Xd != zero_zero(real) )
         => ( real_Vector_of_real(A,aa(real,real,inverse_inverse(real),Xd)) = aa(A,A,inverse_inverse(A),real_Vector_of_real(A,Xd)) ) ) ) ).

% nonzero_of_real_inverse
tff(fact_4822_field__class_Ofield__inverse__zero,axiom,
    ! [A: $tType] :
      ( field(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% field_class.field_inverse_zero
tff(fact_4823_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
         => ( A2 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_4824_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A2 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A2 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_4825_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_4826_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_4827_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A2)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A2)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_4828_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,Xd: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( Xd != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),Xd)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_4829_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,Xd: A] :
          ( aa(real,$o,ord_less_eq(real,R2),real_V7770717601297561774m_norm(A,Xd))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xd))),aa(real,real,inverse_inverse(real),R2)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_4830_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_4831_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_4832_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_4833_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_4834_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_4835_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_4836_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_4837_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_4838_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_4839_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_4840_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_4841_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_4842_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ).

% inverse_eq_divide
tff(fact_4843_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_4844_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: 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_4845_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_4846_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xd: 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),Xd),M)),aa(A,A,inverse_inverse(A),Xd)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xd)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),M)) ) ).

% power_mult_inverse_distrib
tff(fact_4847_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xd: 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),Xd),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xd)),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),Xd)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_4848_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: nat,Xd: 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))),Xd) = aa(A,A,aa(A,fun(A,A),times_times(A),Xd),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xaa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_4849_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( abs_abs(A,aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),abs_abs(A,A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_4850_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: int,Xd: 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))),Xd) = aa(A,A,aa(A,fun(A,A),times_times(A),Xd),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xaa))) ) ).

% mult_inverse_of_int_commute
tff(fact_4851_divide__real__def,axiom,
    ! [Xd: real,Y: real] : divide_divide(real,Xd,Y) = aa(real,real,aa(real,fun(real,real),times_times(real),Xd),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_4852_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_4853_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => aa(A,$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_4854_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4855_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$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_4856_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),Xd)),one_one(A))
        <=> ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
            | aa(A,$o,ord_less_eq(A,one_one(A)),Xd) ) ) ) ).

% inverse_le_1_iff
tff(fact_4857_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_4858_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),Xd))
        <=> ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
            & aa(A,$o,ord_less(A,Xd),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_4859_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_4860_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_4861_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_4862_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_4863_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_4864_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_4865_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa(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_4866_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_4867_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less_eq(A,A2),one_one(A))
           => aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_le_inverse
tff(fact_4868_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),Xd)),one_one(A))
        <=> ( aa(A,$o,ord_less_eq(A,Xd),zero_zero(A))
            | aa(A,$o,ord_less(A,one_one(A)),Xd) ) ) ) ).

% inverse_less_1_iff
tff(fact_4869_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),Xd))
        <=> ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
            & aa(A,$o,ord_less_eq(A,Xd),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4870_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$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,ord_less_eq(A,B2),A2) )
            & ( aa(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_eq(A,A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_4871_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,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,ord_less(A,A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_4872_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_4873_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => ? [N: 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,N)))),Xd) ) ) ).

% reals_Archimedean
tff(fact_4874_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,Xd: 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),Xd)),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)) = Xd ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_4875_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Xd: 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),Xd)),C2) = Y )
          <=> ( Xd = 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_4876_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,$o,ord_less_eq(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,ord_less_eq(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_4877_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,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_4878_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,$o,ord_less_eq(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_4879_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,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,ord_less_eq(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_4880_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_4881_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_4882_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_4883_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_4884_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D2: real,E2: real] :
          ( aa(real,$o,ord_less(real,D2),E2)
         => ( aa(real,$o,P,D2)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_4885_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D2: real,E2: real] :
          ( aa(real,$o,ord_less(real,D2),E2)
         => ( aa(real,$o,P,D2)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono
tff(fact_4886_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),E)
    <=> ? [N6: nat] :
          ( ( N6 != zero_zero(nat) )
          & aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N6)))
          & aa(real,$o,ord_less(real,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N6))),E) ) ) ).

% real_arch_inverse
tff(fact_4887_sqrt__divide__self__eq,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( divide_divide(real,aa(real,real,sqrt,Xd),Xd) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xd)) ) ) ).

% sqrt_divide_self_eq
tff(fact_4888_ln__inverse,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xd)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xd)) ) ) ).

% ln_inverse
tff(fact_4889_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : summable(A,aTP_Lamp_ji(A,fun(nat,A),Xd)) ) ).

% summable_exp
tff(fact_4890_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : summable(A,aTP_Lamp_jj(A,fun(nat,A),Xd)) ) ).

% summable_exp_generic
tff(fact_4891_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xd)
         => ? [N: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
              & aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),Xd) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_4892_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xd: A,M: nat,Nb: nat] :
          ( ( Xd != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,M),Nb)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xd),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),Xd),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xd)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4893_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,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,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_4894_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,$o,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,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_4895_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,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,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_4896_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,$o,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,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_4897_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_4898_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_4899_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_4900_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_4901_log__inverse,axiom,
    ! [A2: real,Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),Xd)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),Xd)) ) ) ) ) ).

% log_inverse
tff(fact_4902_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sums(A,aTP_Lamp_jj(A,fun(nat,A),Xd),aa(A,A,exp(A),Xd)) ) ).

% exp_converges
tff(fact_4903_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : aa(A,A,exp(A),X4) = suminf(A,aTP_Lamp_jj(A,fun(nat,A),X4)) ) ).

% exp_def
tff(fact_4904_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : summable(real,aTP_Lamp_jk(A,fun(nat,real),Xd)) ) ).

% summable_norm_exp
tff(fact_4905_exp__plus__inverse__exp,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),Xd)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xd)))) ).

% exp_plus_inverse_exp
tff(fact_4906_plus__inverse__ge__2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,inverse_inverse(real),Xd))) ) ).

% plus_inverse_ge_2
tff(fact_4907_real__inv__sqrt__pow2,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xd))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xd) ) ) ).

% real_inv_sqrt_pow2
tff(fact_4908_tan__cot,axiom,
    ! [Xd: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),Xd)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),Xd)) ).

% tan_cot
tff(fact_4909_real__le__x__sinh,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,Xd),divide_divide(real,aa(real,real,minus_minus(real,aa(real,real,exp(real),Xd)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xd))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_4910_real__le__abs__sinh,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),abs_abs(real,divide_divide(real,aa(real,real,minus_minus(real,aa(real,real,exp(real),Xd)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xd))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_4911_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xd) )
         => ( 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),Xd),Y)),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jl(A,fun(A,fun(nat,fun(nat,A))),Xd),Y),Nb)),set_ord_atMost(nat,Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4912_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : aa(A,A,exp(A),Xd) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_jm(A,fun(nat,A),Xd))) ) ).

% exp_first_term
tff(fact_4913_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xd)),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,Xd))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_4914_powr__real__of__int,axiom,
    ! [Xd: real,Nb: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( powr(real,Xd,aa(int,real,ring_1_of_int(real),Nb)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),nat2(Nb)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),nat2(aa(int,int,uminus_uminus(int),Nb))))) ) ) ).

% powr_real_of_int
tff(fact_4915_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A,K: nat] : aa(A,A,exp(A),Xd) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jj(A,fun(nat,A),Xd)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_jn(A,fun(nat,fun(nat,A)),Xd),K))) ) ).

% exp_first_terms
tff(fact_4916_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sums(A,aTP_Lamp_jo(A,fun(nat,A),Xd),sinh(A,Xd)) ) ).

% sinh_converges
tff(fact_4917_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sums(A,aTP_Lamp_jp(A,fun(nat,A),Xd),cosh(A,Xd)) ) ).

% cosh_converges
tff(fact_4918_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T6: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
           => ( aa(real,$o,ord_less(real,T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T6),sin(real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_4919_sinh__real__zero__iff,axiom,
    ! [Xd: real] :
      ( ( sinh(real,Xd) = zero_zero(real) )
    <=> ( Xd = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_4920_sinh__real__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less(real,sinh(real,Xd)),sinh(real,Y))
    <=> aa(real,$o,ord_less(real,Xd),Y) ) ).

% sinh_real_less_iff
tff(fact_4921_sinh__real__neg__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,sinh(real,Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xd),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_4922_sinh__real__pos__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),sinh(real,Xd))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xd) ) ).

% sinh_real_pos_iff
tff(fact_4923_sinh__real__nonneg__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),sinh(real,Xd))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd) ) ).

% sinh_real_nonneg_iff
tff(fact_4924_sinh__real__nonpos__iff,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,sinh(real,Xd)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xd),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_4925_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_4926_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_4927_norm__cos__sin,axiom,
    ! [T: real] : real_V7770717601297561774m_norm(complex,complex2(cos(real,T),sin(real,T))) = one_one(real) ).

% norm_cos_sin
tff(fact_4928_complex__scaleR,axiom,
    ! [R2: real,A2: real,B2: real] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),B2)) ).

% complex_scaleR
tff(fact_4929_cosh__real__nonzero,axiom,
    ! [Xd: real] : cosh(real,Xd) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_4930_complex__of__real__def,axiom,
    ! [R2: real] : real_Vector_of_real(complex,R2) = complex2(R2,zero_zero(real)) ).

% complex_of_real_def
tff(fact_4931_complex__of__real__code,axiom,
    ! [X4: real] : real_Vector_of_real(complex,X4) = complex2(X4,zero_zero(real)) ).

% complex_of_real_code
tff(fact_4932_complex__eq__cancel__iff2,axiom,
    ! [Xd: real,Y: real,Xaa: real] :
      ( ( complex2(Xd,Y) = real_Vector_of_real(complex,Xaa) )
    <=> ( ( Xd = Xaa )
        & ( Y = zero_zero(real) ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_4933_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,Xd)),cosh(A,Xd)) = aa(A,A,exp(A),Xd) ) ).

% sinh_plus_cosh
tff(fact_4934_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,Xd)),sinh(A,Xd)) = aa(A,A,exp(A),Xd) ) ).

% cosh_plus_sinh
tff(fact_4935_sinh__less__cosh__real,axiom,
    ! [Xd: real] : aa(real,$o,ord_less(real,sinh(real,Xd)),cosh(real,Xd)) ).

% sinh_less_cosh_real
tff(fact_4936_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : cosh(A,aa(A,A,minus_minus(A,Xd),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xd)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xd)),sinh(A,Y))) ) ).

% cosh_diff
tff(fact_4937_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : sinh(A,aa(A,A,minus_minus(A,Xd),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xd)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xd)),sinh(A,Y))) ) ).

% sinh_diff
tff(fact_4938_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xd: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),complex2(Xd,Y)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_4939_Complex__mult__complex__of__real,axiom,
    ! [Xd: real,Y: real,R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(Xd,Y)),real_Vector_of_real(complex,R2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),Xd),R2),aa(real,real,aa(real,fun(real,real),times_times(real),Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_4940_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),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,Xd)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xd)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_4941_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),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,Xd)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xd)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_4942_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(A,A,tanh(A),Xd) = divide_divide(A,sinh(A,Xd),cosh(A,Xd)) ) ).

% tanh_def
tff(fact_4943_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_4944_cosh__real__pos,axiom,
    ! [Xd: real] : aa(real,$o,ord_less(real,zero_zero(real)),cosh(real,Xd)) ).

% cosh_real_pos
tff(fact_4945_cosh__real__nonpos__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,Y),zero_zero(real))
       => ( aa(real,$o,ord_less_eq(real,cosh(real,Xd)),cosh(real,Y))
        <=> aa(real,$o,ord_less_eq(real,Y),Xd) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_4946_cosh__real__nonneg__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,cosh(real,Xd)),cosh(real,Y))
        <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_4947_cosh__real__nonneg,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),cosh(real,Xd)) ).

% cosh_real_nonneg
tff(fact_4948_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)) = 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,Xd))),cosh(A,Xd)) ) ).

% sinh_double
tff(fact_4949_cosh__real__ge__1,axiom,
    ! [Xd: real] : aa(real,$o,ord_less_eq(real,one_one(real)),cosh(real,Xd)) ).

% cosh_real_ge_1
tff(fact_4950_Complex__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = zero_zero(complex) )
    <=> ( ( A2 = zero_zero(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_0
tff(fact_4951_zero__complex_Ocode,axiom,
    zero_zero(complex) = complex2(zero_zero(real),zero_zero(real)) ).

% zero_complex.code
tff(fact_4952_cosh__real__strict__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),Y)
       => aa(real,$o,ord_less(real,cosh(real,Xd)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_4953_cosh__real__nonneg__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,cosh(real,Xd)),cosh(real,Y))
        <=> aa(real,$o,ord_less(real,Xd),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_4954_cosh__real__nonpos__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,Y),zero_zero(real))
       => ( aa(real,$o,ord_less(real,cosh(real,Xd)),cosh(real,Y))
        <=> aa(real,$o,ord_less(real,Y),Xd) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_4955_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_4956_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),complex2(C2,D)) = 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),D)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),D)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C2))) ).

% complex_mult
tff(fact_4957_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xd)),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,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_4958_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_4959_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xd)),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,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_4960_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_4961_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_4962_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(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))))),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_4963_arcosh__cosh__real,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,real,arcosh(real),cosh(real,Xd)) = Xd ) ) ).

% arcosh_cosh_real
tff(fact_4964_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xd)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_4965_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),S: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_jq(fun(A,real),fun(A,complex),F2)),S) = complex2(aa(set(A),real,groups7311177749621191930dd_sum(A,real,F2),S),zero_zero(real)) ).

% Complex_sum'
tff(fact_4966_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_4967_complex__norm,axiom,
    ! [Xd: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(Xd,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),Xd),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_4968_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,Y: A] :
          ( ( cosh(A,Xd) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),Xd)),aa(A,A,tanh(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),Xd)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_4969_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( sinh(A,Xd) = zero_zero(A) )
        <=> member(A,aa(A,A,exp(A),Xd),insert(A,one_one(A),insert(A,aa(A,A,uminus_uminus(A),one_one(A)),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_4970_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cosh_field_def
tff(fact_4971_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : sinh(A,Z) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sinh_field_def
tff(fact_4972_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( cosh(A,Xd) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_4973_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : cosh(A,Xd) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Xd)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xd)))) ) ).

% cosh_def
tff(fact_4974_cosh__ln__real,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( cosh(real,aa(real,real,ln_ln(real),Xd)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,inverse_inverse(real),Xd)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_4975_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xd: A] : sinh(A,Xd) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,minus_minus(A,aa(A,A,exp(A),Xd)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xd)))) ) ).

% sinh_def
tff(fact_4976_sinh__ln__real,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( sinh(real,aa(real,real,ln_ln(real),Xd)) = divide_divide(real,aa(real,real,minus_minus(real,Xd),aa(real,real,inverse_inverse(real),Xd)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_4977_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)) = 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_4978_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A),Xd: A] :
          ( aa(nat,$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,Xd,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_jr(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I),Xs),A2),Xd)) ) ) ).

% upd_rule
tff(fact_4979_cot__less__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),zero_zero(real))
       => aa(real,$o,ord_less(real,aa(real,real,cot(real),Xd)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_4980_cot__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,cot(A),zero_zero(A)) = zero_zero(A) ) ) ).

% cot_zero
tff(fact_4981_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_4982_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( ( bit_se2239418461657761734s_mask(A,Nb) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_4983_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_4984_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_4985_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_4986_cot__npi,axiom,
    ! [Nb: nat] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_4987_cot__periodic,axiom,
    ! [Xd: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,cot(real),Xd) ).

% cot_periodic
tff(fact_4988_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,Nb)) = bit_se2239418461657761734s_mask(A,Nb) ) ).

% of_nat_mask_eq
tff(fact_4989_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,Nb)) = bit_se2239418461657761734s_mask(A,Nb) ) ).

% of_int_mask_eq
tff(fact_4990_ex__mask__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ? [X: nat] : bit_se2239418461657761734s_mask(word(A),X) = one_one(word(A)) ) ).

% ex_mask_1
tff(fact_4991_mask__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,one_one(nat)) = one_one(A) ) ) ).

% mask_1
tff(fact_4992_take__bit__eq__mask,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),bit_se5824344872417868541ns_and(A),A2),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% take_bit_eq_mask
tff(fact_4993_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_4994_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_4995_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_4996_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_4997_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: A] : aa(A,A,cot(A),X4) = divide_divide(A,cos(A,X4),sin(A,X4)) ) ).

% cot_def
tff(fact_4998_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_4999_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_5000_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_5001_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(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)),Xd),bit_se2239418461657761734s_mask(word(A),Nb)) = Xd ) ) ) ).

% less_mask_eq
tff(fact_5002_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_5003_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_5004_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_5005_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_5006_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_5007_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_5008_add__mask__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)),Xd),aa(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)),Xd),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% add_mask_fold
tff(fact_5009_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_5010_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)),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_5011_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_5012_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_5013_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),Xd))
         => 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)),Xd),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_5014_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),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_5015_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A)] :
          ( aa(word(A),$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))
         => ( modulo_modulo(word(A),Xd,aa(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)),Xd),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_5016_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(word(A),nat,size_size(word(A)),W))
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),M)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% word_unat_mask_lt
tff(fact_5017_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)) = 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_5018_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_5019_cot__gt__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,cot(real),Xd)) ) ) ).

% cot_gt_zero
tff(fact_5020_tan__cot_H,axiom,
    ! [Xd: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),Xd)) = aa(real,real,cot(real),Xd) ).

% tan_cot'
tff(fact_5021_exp__two__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_5022_exp__two__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),bit0(one2))),real_Vector_of_real(complex,pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_5023_VEBTi_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Nodei(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_array(vEBT_VEBTi,size_size(vEBT_VEBTi),X13)),aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size(3)
tff(fact_5024_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_5025_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_5026_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_5027_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_5028_exp__pi__i,axiom,
    aa(complex,complex,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_5029_exp__pi__i_H,axiom,
    aa(complex,complex,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_5030_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_5031_less__eq__mask,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less_eq(nat,Nb),bit_se2239418461657761734s_mask(nat,Nb)) ).

% less_eq_mask
tff(fact_5032_nat__mask__eq,axiom,
    ! [Nb: nat] : nat2(bit_se2239418461657761734s_mask(int,Nb)) = bit_se2239418461657761734s_mask(nat,Nb) ).

% nat_mask_eq
tff(fact_5033_complex__i__not__one,axiom,
    imaginary_unit != one_one(complex) ).

% complex_i_not_one
tff(fact_5034_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero(complex) ).

% complex_i_not_zero
tff(fact_5035_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se2239418461657761734s_mask(int,Nb)) ).

% mask_nonnegative_int
tff(fact_5036_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,ord_less(int,bit_se2239418461657761734s_mask(int,Nb)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_5037_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_5038_Complex__eq__i,axiom,
    ! [Xd: real,Y: real] :
      ( ( complex2(Xd,Y) = imaginary_unit )
    <=> ( ( Xd = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_5039_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_5040_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_5041_i__complex__of__real,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,R2)) = complex2(zero_zero(real),R2) ).

% i_complex_of_real
tff(fact_5042_complex__of__real__i,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),imaginary_unit) = complex2(zero_zero(real),R2) ).

% complex_of_real_i
tff(fact_5043_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_5044_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_5045_mask__half__int,axiom,
    ! [Nb: nat] : 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_5046_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_5047_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_5048_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_5049_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_5050_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Nodei(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_array(vEBT_VEBTi,vEBT_size_VEBTi,X13)),aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size_gen(1)
tff(fact_5051_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_minus_ii
tff(fact_5052_csqrt__ii,axiom,
    csqrt(imaginary_unit) = 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_5053_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% csqrt_eq_0
tff(fact_5054_csqrt__0,axiom,
    csqrt(zero_zero(complex)) = zero_zero(complex) ).

% csqrt_0
tff(fact_5055_csqrt__1,axiom,
    csqrt(one_one(complex)) = one_one(complex) ).

% csqrt_1
tff(fact_5056_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% csqrt_eq_1
tff(fact_5057_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_5058_Arg__ii,axiom,
    arg(imaginary_unit) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_ii
tff(fact_5059_Arg__zero,axiom,
    arg(zero_zero(complex)) = zero_zero(real) ).

% Arg_zero
tff(fact_5060_of__real__sqrt,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( real_Vector_of_real(complex,aa(real,real,sqrt,Xd)) = csqrt(real_Vector_of_real(complex,Xd)) ) ) ).

% of_real_sqrt
tff(fact_5061_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,ord_less_eq(real,arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_5062_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(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_5063_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),bit0(one2)) = aa(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_5064_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xd: fun(A,nat),X2: A] : size_option(A,Xd,aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xd,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_5065_norm__cis,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,cis(A2)) = one_one(real) ).

% norm_cis
tff(fact_5066_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_5067_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_5068_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_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(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_cis__neq__zero,axiom,
    ! [A2: real] : cis(A2) != zero_zero(complex) ).

% cis_neq_zero
tff(fact_5073_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_5074_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(bit0(Nb)) = aa(num,num,bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_5075_semiring__norm_I28_J,axiom,
    ! [Nb: num] : bitM(aa(num,num,bit1,Nb)) = aa(num,num,bit1,bit0(Nb)) ).

% semiring_norm(28)
tff(fact_5076_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = bit0(Nb) ).

% inc_BitM_eq
tff(fact_5077_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = aa(num,num,bit1,Nb) ).

% BitM_inc_eq
tff(fact_5078_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( cis(arg(Z)) = sgn_sgn(complex,Z) ) ) ).

% cis_Arg
tff(fact_5079_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_5080_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_5081_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_5082_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_5083_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_5084_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_5085_cis__Arg__unique,axiom,
    ! [Z: complex,Xd: real] :
      ( ( sgn_sgn(complex,Z) = cis(Xd) )
     => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Xd)
       => ( aa(real,$o,ord_less_eq(real,Xd),pi)
         => ( arg(Z) = Xd ) ) ) ) ).

% cis_Arg_unique
tff(fact_5086_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( sgn_sgn(complex,Z) = cis(arg(Z)) )
        & aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,ord_less_eq(real,arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_5087_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xd: fun(A,nat)] : size_option(A,Xd,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_5088_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => bij_betw(nat,complex,aTP_Lamp_js(nat,fun(nat,complex),Nb),set_ord_lessThan(nat,Nb),collect(complex,aTP_Lamp_gb(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_5089_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_jt(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_5090_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,I6: set(nat),A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) )
            & aa(set(nat),$o,ord_less_eq(set(nat),I6),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_ju(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),A3),Xs),Xsi),one_one(assn),I6)) ).

% listI_assn_def
tff(fact_5091_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_jv(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Nb)) ) ).

% divmod_algorithm_code(5)
tff(fact_5092_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S3: set(A),T4: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S3),S4),aa(set(B),set(B),minus_minus(set(B),T4),T3))
             => ( ! [A5: A] :
                    ( member(A,A5,S4)
                   => ( aa(B,C,G,aa(A,B,H,A5)) = zero_zero(C) ) )
               => ( ! [B3: B] :
                      ( member(B,B3,T3)
                     => ( 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_jw(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T4) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_5093_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S3: set(A),T4: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S3),S4),aa(set(B),set(B),minus_minus(set(B),T4),T3))
             => ( ! [A5: A] :
                    ( member(A,A5,S4)
                   => ( aa(B,C,G,aa(A,B,H,A5)) = one_one(C) ) )
               => ( ! [B3: B] :
                      ( member(B,B3,T3)
                     => ( aa(B,C,G,B3) = one_one(C) ) )
                 => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_jx(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S3) = groups7121269368397514597t_prod(B,C,G,T4) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_5094_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_jy(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_5095_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_jz(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_5096_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_ka(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_5097_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_kb(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_5098_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),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_kc(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_5099_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_kd(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_5100_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_ke(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_5101_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A3) )
     => ( ? [X_13: A] : aa(A,$o,P,X_13)
      <=> (A3) ) ) ).

% verit_sko_ex'
tff(fact_5102_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_13: A] : aa(A,$o,P,X_13)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_kf(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_5103_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_kf(fun(A,$o),fun(A,$o),P)))
      <=> (A3) )
     => ( ! [X_13: A] : aa(A,$o,P,X_13)
      <=> (A3) ) ) ).

% verit_sko_forall'
tff(fact_5104_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B4: A,A3: A,P: fun(A,$o)] :
      ( ( B4 = A3 )
     => ( ( fChoice(A,P) = A3 )
      <=> ( fChoice(A,P) = B4 ) ) ) ).

% verit_sko_forall''
tff(fact_5105_verit__sko__ex__indirect,axiom,
    ! [A: $tType,Xd: A,P: fun(A,$o)] :
      ( ( Xd = fChoice(A,P) )
     => ( ? [X_13: A] : aa(A,$o,P,X_13)
      <=> aa(A,$o,P,Xd) ) ) ).

% verit_sko_ex_indirect
tff(fact_5106_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,Xd: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( Xd = fChoice(A,P) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
          <=> aa(A,$o,P2,X) )
       => ( ? [X_13: A] : aa(A,$o,P2,X_13)
        <=> aa(A,$o,P,Xd) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_5107_verit__sko__forall__indirect,axiom,
    ! [A: $tType,Xd: A,P: fun(A,$o)] :
      ( ( Xd = fChoice(A,aTP_Lamp_kf(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_13: A] : aa(A,$o,P,X_13)
      <=> aa(A,$o,P,Xd) ) ) ).

% verit_sko_forall_indirect
tff(fact_5108_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,Xd: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( Xd = fChoice(A,aTP_Lamp_kf(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
          <=> aa(A,$o,P2,X) )
       => ( ! [X_13: A] : aa(A,$o,P2,X_13)
        <=> aa(A,$o,P,Xd) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_5109_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_kg(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ki(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_5110_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_kg(nat,fun(nat,fun(nat,$o)),Nb)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_kk(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_5111_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_kl(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ki(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ).

% sum.triangle_reindex
tff(fact_5112_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_kl(nat,fun(nat,fun(nat,$o)),Nb)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_kk(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,Nb)) ) ).

% prod.triangle_reindex
tff(fact_5113_ln__neg__is__const,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,Xd),zero_zero(real))
     => ( aa(real,real,ln_ln(real),Xd) = the(real,aTP_Lamp_km(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_5114_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_kn(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_5115_divmod__nat__def,axiom,
    ! [M: nat,Nb: nat] : divmod_nat(M,Nb) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,divide_divide(nat,M,Nb)),modulo_modulo(nat,M,Nb)) ).

% divmod_nat_def
tff(fact_5116_pi__half,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))) = the(real,aTP_Lamp_ko(real,$o)) ).

% pi_half
tff(fact_5117_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_ko(real,$o))) ).

% pi_def
tff(fact_5118_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_kp(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_5119_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_kq(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_5120_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(Nb),real_V7770717601297561774m_norm(complex,C2)))),cis(divide_divide(real,arg(C2),aa(nat,real,semiring_1_of_nat(real),Nb))))),collect(complex,aTP_Lamp_gb(nat,fun(complex,$o),Nb)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_kr(complex,fun(nat,fun(complex,$o)),C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_5121_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)),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_5122_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)) = Xd ) ).

% bit.double_compl
tff(fact_5123_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xd) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( Xd = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_5124_real__root__zero,axiom,
    ! [Nb: nat] : aa(real,real,root(Nb),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_5125_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),Xd) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_5126_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_5127_real__root__Suc__0,axiom,
    ! [Xd: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xd) = Xd ).

% real_root_Suc_0
tff(fact_5128_real__root__eq__iff,axiom,
    ! [Nb: nat,Xd: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xd) = aa(real,real,root(Nb),Y) )
      <=> ( Xd = Y ) ) ) ).

% real_root_eq_iff
tff(fact_5129_root__0,axiom,
    ! [Xd: real] : aa(real,real,root(zero_zero(nat)),Xd) = zero_zero(real) ).

% root_0
tff(fact_5130_and__and__not,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)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),B2)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),B2)) = zero_zero(word(A)) ) ).

% and_and_not
tff(fact_5131_word__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)) = zero_zero(word(A)) ) ).

% word_and_not
tff(fact_5132_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_5133_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_5134_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xd) = zero_zero(real) )
      <=> ( Xd = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_5135_real__root__less__iff,axiom,
    ! [Nb: nat,Xd: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xd)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,ord_less(real,Xd),Y) ) ) ).

% real_root_less_iff
tff(fact_5136_NOT__mask__AND__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [W: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),bit_se2239418461657761734s_mask(A,Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) = zero_zero(A) ) ).

% NOT_mask_AND_mask
tff(fact_5137_real__root__le__iff,axiom,
    ! [Nb: nat,Xd: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(Nb),Xd)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,ord_less_eq(real,Xd),Y) ) ) ).

% real_root_le_iff
tff(fact_5138_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_5139_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xd) = one_one(real) )
      <=> ( Xd = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_5140_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_5141_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_5142_word__add__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_add_not
tff(fact_5143_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_5144_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_5145_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xd)),zero_zero(real))
      <=> aa(real,$o,ord_less(real,Xd),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_5146_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_5147_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(Nb),Xd)),zero_zero(real))
      <=> aa(real,$o,ord_less_eq(real,Xd),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_5148_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_5149_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xd)),one_one(real))
      <=> aa(real,$o,ord_less(real,Xd),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_5150_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,ord_less_eq(real,one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_5151_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(Nb),Xd)),one_one(real))
      <=> aa(real,$o,ord_less_eq(real,Xd),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_5152_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_5153_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),Nb) = Xd ) ) ) ).

% real_root_pow_pos2
tff(fact_5154_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_5155_real__root__commute,axiom,
    ! [M: nat,Nb: nat,Xd: real] : aa(real,real,root(M),aa(real,real,root(Nb),Xd)) = aa(real,real,root(Nb),aa(real,real,root(M),Xd)) ).

% real_root_commute
tff(fact_5156_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K)) ) ).

% of_int_not_eq
tff(fact_5157_take__bit__not__take__bit,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),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) ) ).

% take_bit_not_take_bit
tff(fact_5158_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_5159_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_5160_real__root__mult__exp,axiom,
    ! [M: nat,Nb: nat,Xd: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)),Xd) = aa(real,real,root(M),aa(real,real,root(Nb),Xd)) ).

% real_root_mult_exp
tff(fact_5161_real__root__mult,axiom,
    ! [Nb: nat,Xd: real,Y: real] : aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),times_times(real),Xd),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(Nb),Xd)),aa(real,real,root(Nb),Y)) ).

% real_root_mult
tff(fact_5162_real__root__minus,axiom,
    ! [Nb: nat,Xd: real] : aa(real,real,root(Nb),aa(real,real,uminus_uminus(real),Xd)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Nb),Xd)) ).

% real_root_minus
tff(fact_5163_real__root__inverse,axiom,
    ! [Nb: nat,Xd: real] : aa(real,real,root(Nb),aa(real,real,inverse_inverse(real),Xd)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Nb),Xd)) ).

% real_root_inverse
tff(fact_5164_real__root__divide,axiom,
    ! [Nb: nat,Xd: real,Y: real] : aa(real,real,root(Nb),divide_divide(real,Xd,Y)) = divide_divide(real,aa(real,real,root(Nb),Xd),aa(real,real,root(Nb),Y)) ).

% real_root_divide
tff(fact_5165_real__root__pos__pos__le,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(Nb),Xd)) ) ).

% real_root_pos_pos_le
tff(fact_5166_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_5167_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_5168_mask__eq__0__eq__x,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),W) = zero_zero(word(A)) )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = Xd ) ) ) ).

% mask_eq_0_eq_x
tff(fact_5169_mask__eq__x__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),W) = Xd )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = zero_zero(word(A)) ) ) ) ).

% mask_eq_x_eq_0
tff(fact_5170_real__root__less__mono,axiom,
    ! [Nb: nat,Xd: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,Xd),Y)
       => aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xd)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_5171_real__root__le__mono,axiom,
    ! [Nb: nat,Xd: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,Xd),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,root(Nb),Xd)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_5172_real__root__power,axiom,
    ! [Nb: nat,Xd: 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),Xd),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),K) ) ) ).

% real_root_power
tff(fact_5173_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_5174_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_5175_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_5176_real__root__abs,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),abs_abs(real,Xd)) = abs_abs(real,aa(real,real,root(Nb),Xd)) ) ) ).

% real_root_abs
tff(fact_5177_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_5178_NOT__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),Xd)),one_one(word(A))) ) ).

% NOT_eq
tff(fact_5179_mask__AND__NOT__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)),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)))),bit_se2239418461657761734s_mask(word(A),Nb)) = zero_zero(word(A)) ) ).

% mask_AND_NOT_mask
tff(fact_5180_sgn__root,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( sgn_sgn(real,aa(real,real,root(Nb),Xd)) = sgn_sgn(real,Xd) ) ) ).

% sgn_root
tff(fact_5181_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_5182_real__root__gt__zero,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,root(Nb),Xd)) ) ) ).

% real_root_gt_zero
tff(fact_5183_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,Nb),N2)
       => ( aa(real,$o,ord_less(real,one_one(real)),Xd)
         => aa(real,$o,ord_less(real,aa(real,real,root(N2),Xd)),aa(real,real,root(Nb),Xd)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_5184_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_5185_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_5186_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( abs_abs(real,aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb))) = abs_abs(real,Y) ) ) ).

% root_abs_power
tff(fact_5187_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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_5188_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_5189_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,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_5190_real__root__pos__pos,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(Nb),Xd)) ) ) ).

% real_root_pos_pos
tff(fact_5191_real__root__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,Nb),N2)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
         => ( aa(real,$o,ord_less(real,Xd),one_one(real))
           => aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xd)),aa(real,real,root(N2),Xd)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_5192_real__root__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,Nb),N2)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),Xd)
         => aa(real,$o,ord_less_eq(real,aa(real,real,root(N2),Xd)),aa(real,real,root(Nb),Xd)) ) ) ) ).

% real_root_decreasing
tff(fact_5193_odd__real__root__pow,axiom,
    ! [Nb: nat,Xd: 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),Xd)),Nb) = Xd ) ) ).

% odd_real_root_pow
tff(fact_5194_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xd: 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) = Xd )
       => ( aa(real,real,root(Nb),Xd) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_5195_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xd: 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),Xd),Nb)) = Xd ) ) ).

% odd_real_root_power_cancel
tff(fact_5196_real__root__pow__pos,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),Nb) = Xd ) ) ) ).

% real_root_pow_pos
tff(fact_5197_real__root__power__cancel,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
       => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb)) = Xd ) ) ) ).

% real_root_power_cancel
tff(fact_5198_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb) = Xd )
         => ( aa(real,real,root(Nb),Xd) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_5199_real__root__increasing,axiom,
    ! [Nb: nat,N2: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,Nb),N2)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
         => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
           => aa(real,$o,ord_less_eq(real,aa(real,real,root(Nb),Xd)),aa(real,real,root(N2),Xd)) ) ) ) ) ).

% real_root_increasing
tff(fact_5200_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_5201_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_5202_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),sgn_sgn(real,Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_5203_sgn__power__root,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,aa(real,real,root(Nb),Xd))),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,aa(real,real,root(Nb),Xd))),Nb)) = Xd ) ) ).

% sgn_power_root
tff(fact_5204_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)
       => ( aa(real,real,log(B2),aa(real,real,root(Nb),A2)) = divide_divide(real,aa(real,real,log(B2),A2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_root
tff(fact_5205_log__base__root,axiom,
    ! [Nb: nat,B2: real,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
       => ( aa(real,real,log(aa(real,real,root(Nb),B2)),Xd) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Xd)) ) ) ) ).

% log_base_root
tff(fact_5206_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)) = divide_divide(real,aa(real,real,ln_ln(real),B2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% ln_root
tff(fact_5207_split__root,axiom,
    ! [P: fun(real,$o),Nb: nat,Xd: real] :
      ( aa(real,$o,P,aa(real,real,root(Nb),Xd))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y5)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,Y5)),Nb)) = Xd )
             => aa(real,$o,P,Y5) ) ) ) ) ).

% split_root
tff(fact_5208_root__powr__inverse,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => ( aa(real,real,root(Nb),Xd) = powr(real,Xd,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_5209_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)),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_5210_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,bit_se5641148757651400278ts_bit(int,K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5211_neg__mask__add__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)),Xd),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)),Xd),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% neg_mask_add_mask
tff(fact_5212_new__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xd: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,Nb,Xd),aa(A,fun(array(A),assn),aTP_Lamp_ks(nat,fun(A,fun(array(A),assn)),Nb),Xd)) ) ).

% new_rule
tff(fact_5213_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.right_idem
tff(fact_5214_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.left_idem
tff(fact_5215_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),A2) = A2 ) ).

% or.idem
tff(fact_5216_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),zero_zero(A)) = A2 ) ).

% or.right_neutral
tff(fact_5217_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A2) = A2 ) ).

% or.left_neutral
tff(fact_5218_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_or
tff(fact_5219_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_5220_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),Xd) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_5221_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_5222_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_5223_word__bitwise__m1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)))),Xd) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_bitwise_m1_simps(4)
tff(fact_5224_word__or__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),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_5225_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(M))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),Nb) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5226_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),Nb) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5227_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xd))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xd)) ) ).

% or_numerals(8)
tff(fact_5228_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_5229_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),Xd) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_5230_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_5231_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_5232_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_5233_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_5234_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_5235_word__or__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_or_not
tff(fact_5236_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_5237_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_5238_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xd))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xd)) ) ).

% or_numerals(5)
tff(fact_5239_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(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,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_5240_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: num] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(W))),aa(num,nat,numeral_numeral(nat),Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(Nb)) ) ) ).

% bit_numeral_simps(2)
tff(fact_5241_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(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,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_5242_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: num] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(Nb)) ) ) ).

% bit_numeral_simps(3)
tff(fact_5243_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_5244_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$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_5245_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa(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,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_5246_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa(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,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_5247_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(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_5248_bin__nth__minus__Bit0,axiom,
    ! [Nb: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),bit0(W))),Nb)
      <=> aa(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_5249_bin__nth__minus__Bit1,axiom,
    ! [Nb: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,W))),Nb)
      <=> aa(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_5250_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_5251_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_5252_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_5253_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: 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,Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_5254_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: 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,Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_5255_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
         => ( 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,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_5256_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% bit_not_int_iff
tff(fact_5257_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),Nb)
    <=> aa(nat,$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_5258_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% bit_numeral_simps(1)
tff(fact_5259_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,Nb)) ) ).

% not_bit_1_Suc
tff(fact_5260_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( 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_5261_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5262_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(num,nat,numeral_numeral(nat),M)),Nb) ) ) ).

% bit_numeral_iff
tff(fact_5263_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),zero_zero(A)) = Xd ) ).

% bit.disj_zero_right
tff(fact_5264_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_5265_word__log__esimps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),zero_zero(word(A))),Xd) = Xd ) ).

% word_log_esimps(9)
tff(fact_5266_word__log__esimps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),zero_zero(word(A))) = Xd ) ).

% word_log_esimps(3)
tff(fact_5267_word__or__zero,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)),bit_se1065995026697491101ons_or(word(A)),A2),B2) = zero_zero(word(A)) )
        <=> ( ( A2 = zero_zero(word(A)) )
            & ( B2 = zero_zero(word(A)) ) ) ) ) ).

% word_or_zero
tff(fact_5268_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            & ( M != Nb ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_5269_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_5270_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.left_commute
tff(fact_5271_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),A2) ) ).

% or.commute
tff(fact_5272_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_or_iff
tff(fact_5273_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.assoc
tff(fact_5274_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_and_iff
tff(fact_5275_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_5276_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_5277_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)),Xd) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Xd)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xd)) ) ).

% bit.conj_disj_distrib2
tff(fact_5278_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)),Xd) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Xd)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),Xd)) ) ).

% bit.disj_conj_distrib2
tff(fact_5279_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_5280_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(nat,M),Nb) ) ) ).

% bit_of_nat_iff_bit
tff(fact_5281_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_or_eq
tff(fact_5282_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),Nb)
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_5283_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_take_bit_iff
tff(fact_5284_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: $o,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa($o,A,zero_neq_one_of_bool(A),(B2))),Nb)
        <=> ( (B2)
            & ( Nb = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_5285_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_5286_and__eq__not__not__or,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,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% and_eq_not_not_or
tff(fact_5287_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% or_eq_not_not_and
tff(fact_5288_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,bit_se5641148757651400278ts_bit(A,A2),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ).

% signed_take_bit_def
tff(fact_5289_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N)
             => aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N) )
         => ( 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_5290_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_5291_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_5292_word__plus__and__or__coroll,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),Y) = zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),Y) ) ) ) ).

% word_plus_and_or_coroll
tff(fact_5293_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa(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,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% bit_not_int_iff'
tff(fact_5294_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          bit_se8732182000553998342ip_bit(A,Nb,A2) = aa(A,A,
            aa(nat,fun(A,A),
              $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),
              Nb),
            A2) ) ).

% flip_bit_eq_if
tff(fact_5295_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_5296_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xd: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xd) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xd) = 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)) )
               => ( Xd = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_5297_time__array__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xd: A,H: heap_ext(product_unit)] : time_time(array(A),array_new(A,Nb,Xd),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ) ).

% time_array_new
tff(fact_5298_not__int__div__2,axiom,
    ! [K: 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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% not_int_div_2
tff(fact_5299_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_5300_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_5301_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_5302_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,M: nat,K: int] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
     => ( aa(nat,$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_5303_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,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5304_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),Nb) ) ) ).

% bit_Suc
tff(fact_5305_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(nat,$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_5306_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) )
         => ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_5307_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xd) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_5308_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N),M3)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M3)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) )
         => ~ ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,N),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_5309_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_5310_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_5311_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_5312_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),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_5313_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,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5314_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_not_iff_eq
tff(fact_5315_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_5316_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_5317_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_5318_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),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_5319_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,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5320_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,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
          <=> ( aa(nat,$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_5321_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_5322_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_5323_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_5324_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [J3: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J3))
         => ( aa(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,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_5325_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(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_5326_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_5327_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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% not_int_rec
tff(fact_5328_Bit__Operations_Oset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,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,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_5329_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,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,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_5330_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,bit_se5641148757651400278ts_bit(int,K),Nb)))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_5331_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_5332_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_5333_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,minus_minus(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_5334_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_5335_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),Xd) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_5336_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),A2) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_5337_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A2) = A2 ) ).

% xor.left_neutral
tff(fact_5338_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),zero_zero(A)) = A2 ) ).

% xor.right_neutral
tff(fact_5339_test__bit__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),one_one(word(A))),Nb)
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% test_bit_1
tff(fact_5340_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_xor
tff(fact_5341_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),Y)) ) ).

% bit.xor_compl_left
tff(fact_5342_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),Y)) ) ).

% bit.xor_compl_right
tff(fact_5343_map__nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(nat)] : aa(list(nat),list($o),map(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_5344_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        & aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_5345_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,ord_less(int,K),zero_zero(int))
        | aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_5346_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
      <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_5347_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,ord_less(int,K),zero_zero(int))
        <=> aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_5348_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_5349_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),Xd) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_5350_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xd) ) ).

% bit.xor_one_right
tff(fact_5351_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),Xd) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xd) ) ).

% bit.xor_one_left
tff(fact_5352_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_5353_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xd))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(Xd)) ) ).

% xor_numerals(8)
tff(fact_5354_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xd))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xd)) ) ).

% xor_numerals(5)
tff(fact_5355_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_5356_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_5357_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_5358_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_5359_or__nat__numerals_I4_J,axiom,
    ! [Xd: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xd))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xd)) ).

% or_nat_numerals(4)
tff(fact_5360_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_5361_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_5362_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: 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,Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_5363_or__nat__numerals_I3_J,axiom,
    ! [Xd: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xd))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xd)) ).

% or_nat_numerals(3)
tff(fact_5364_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_5365_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_5366_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_5367_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_5368_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_5369_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_5370_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_5371_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_5372_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: 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,Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_5373_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xd: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xd))),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),Xd)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_5374_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_5375_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_5376_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_5377_int__or__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),zero_zero(int)) = I ).

% int_or_code(2)
tff(fact_5378_int__or__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),zero_zero(int)),J) = J ).

% int_or_code(1)
tff(fact_5379_nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),zero_zero(word(A))),Nb) ) ).

% nth_0
tff(fact_5380_word__exists__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ? [X_1: nat] : aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),X_1) ) ) ).

% word_exists_nth
tff(fact_5381_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),Nb)
    <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_5382_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        | aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_5383_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),Nb)
        <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_xor_iff
tff(fact_5384_or__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Nb) = nat2(aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% or_nat_def
tff(fact_5385_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_5386_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa(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_5387_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_xor_eq
tff(fact_5388_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xd: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)),Xd) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Xd)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xd)) ) ).

% bit.conj_xor_distrib2
tff(fact_5389_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_5390_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.assoc
tff(fact_5391_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),A2) ) ).

% xor.commute
tff(fact_5392_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.left_commute
tff(fact_5393_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_5394_int__xor__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),zero_zero(int)),J) = J ).

% int_xor_code(1)
tff(fact_5395_int__xor__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),I),zero_zero(int)) = I ).

% int_xor_code(2)
tff(fact_5396_XOR__lower,axiom,
    ! [Xd: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xd),Y)) ) ) ).

% XOR_lower
tff(fact_5397_OR__lower,axiom,
    ! [Xd: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xd),Y)) ) ) ).

% OR_lower
tff(fact_5398_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),L)
     => aa(int,$o,ord_less_eq(int,K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_5399_plus__and__or,axiom,
    ! [Xd: 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),Xd),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xd),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),Y) ).

% plus_and_or
tff(fact_5400_word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),U),N)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),V),N) ) )
         => ( U = V ) ) ) ).

% word_eqI
tff(fact_5401_test__bit__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$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_5402_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_5403_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa(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_5404_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_5405_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A)))),zero_zero(nat))
         => aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),zero_zero(nat)) ) ) ).

% lsb_this_or_next
tff(fact_5406_word__leI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),U),N)
               => aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),V),N) ) )
         => aa(word(A),$o,ord_less_eq(word(A),U),V) ) ) ).

% word_leI
tff(fact_5407_nth__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,I: nat] :
          ( aa(nat,$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_5408_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),Y) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xd),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),Y)) ) ).

% bit.xor_def
tff(fact_5409_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xd: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xd),Y) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xd),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xd)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_5410_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_5411_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_5412_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_5413_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,nat2(K)),Nb)
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ).

% bit_nat_iff
tff(fact_5414_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),one_one(word(A))) = zero_zero(word(A)) )
         => aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),zero_zero(nat)) ) ) ).

% overflow_imp_lsb
tff(fact_5415_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,bit_se5641148757651400278ts_bit(word(A),Nb),zero_zero(nat)),one_one(word(A)),zero_zero(word(A))) ) ).

% word_and_1
tff(fact_5416_test__bit__bin_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),Nb) ) ) ) ).

% test_bit_bin'
tff(fact_5417_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_5418_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_5419_bang__is__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),M)
         => aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),Xd) ) ) ).

% bang_is_le
tff(fact_5420_bit__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,M),Nb)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),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_5421_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_5422_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),zero_zero(nat)) ) ) ).

% odd_iff_lsb
tff(fact_5423_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: nat,Xd: word(A)] :
          ( ( Y = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),Y) != zero_zero(word(A)) )
          <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb) ) ) ) ).

% and_neq_0_is_nth
tff(fact_5424_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),aa(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_5425_XOR__upper,axiom,
    ! [Xd: int,Nb: nat,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less(int,Xd),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),Xd),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_5426_OR__upper,axiom,
    ! [Xd: int,Nb: nat,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
     => ( aa(int,$o,ord_less(int,Xd),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),Xd),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_5427_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_5428_test__bit__int__code_I1_J,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,zero_zero(int)),Nb) ).

% test_bit_int_code(1)
tff(fact_5429_int__and__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),zero_zero(int)) = zero_zero(int) ).

% int_and_code(2)
tff(fact_5430_int__and__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),zero_zero(int)),J) = zero_zero(int) ).

% int_and_code(1)
tff(fact_5431_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_5432_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_5433_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),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_5434_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_5435_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_5436_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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% xor_int_rec
tff(fact_5437_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_5438_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_5439_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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% or_int_rec
tff(fact_5440_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,modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),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),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_5441_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,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% or_int_unfold
tff(fact_5442_Bit__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: $o] : bits_Bit_integer(code_integer_of_int(Xaa),(Xd)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),(Xd))),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_5443_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_5444_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_5445_word__bitwise__m1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: 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)))),Xd) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd) ) ).

% word_bitwise_m1_simps(6)
tff(fact_5446_word__bitwise__m1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd) ) ).

% word_bitwise_m1_simps(7)
tff(fact_5447_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_5448_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_5449_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_5450_xor__nat__numerals_I3_J,axiom,
    ! [Xd: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xd))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xd)) ).

% xor_nat_numerals(3)
tff(fact_5451_xor__nat__numerals_I4_J,axiom,
    ! [Xd: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xd))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xd)) ).

% xor_nat_numerals(4)
tff(fact_5452_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_5453_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_5454_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_5455_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_5456_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_5457_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_5458_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_5459_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_5460_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_5461_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_5462_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_5463_word__log__esimps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),zero_zero(word(A))) = Xd ) ).

% word_log_esimps(5)
tff(fact_5464_word__log__esimps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),zero_zero(word(A))),Xd) = Xd ) ).

% word_log_esimps(11)
tff(fact_5465_word__bw__same_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),Xd) = zero_zero(word(A)) ) ).

% word_bw_same(3)
tff(fact_5466_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_5467_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_5468_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_5469_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_5470_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_5471_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_5472_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,Nb),aa(num,num,bit1,M)) = bitM(bit_or_not_num_neg(Nb,M)) ).

% or_not_num_neg.simps(9)
tff(fact_5473_xor__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Nb) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% xor_nat_def
tff(fact_5474_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_5475_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_5476_or__not__num__neg_Oelims,axiom,
    ! [Xd: num,Xaa: num,Y: num] :
      ( ( bit_or_not_num_neg(Xd,Xaa) = Y )
     => ( ( ( Xd = one2 )
         => ( ( Xaa = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( Xd = one2 )
           => ! [M2: num] :
                ( ( Xaa = bit0(M2) )
               => ( Y != aa(num,num,bit1,M2) ) ) )
         => ( ( ( Xd = one2 )
             => ! [M2: num] :
                  ( ( Xaa = aa(num,num,bit1,M2) )
                 => ( Y != aa(num,num,bit1,M2) ) ) )
           => ( ( ? [N: num] : Xd = bit0(N)
               => ( ( Xaa = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N: num] :
                    ( ( Xd = bit0(N) )
                   => ! [M2: num] :
                        ( ( Xaa = bit0(M2) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M2)) ) ) )
               => ( ! [N: num] :
                      ( ( Xd = bit0(N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa(num,num,bit1,M2) )
                         => ( Y != bit0(bit_or_not_num_neg(N,M2)) ) ) )
                 => ( ( ? [N: num] : Xd = aa(num,num,bit1,N)
                     => ( ( Xaa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N: num] :
                          ( ( Xd = aa(num,num,bit1,N) )
                         => ! [M2: num] :
                              ( ( Xaa = bit0(M2) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M2)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xd = aa(num,num,bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa(num,num,bit1,M2) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_5477_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),Xd))
         => ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),Y)),Nb)
            <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb)
                | aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),Nb) ) )
            & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),Y)),Nb)
            <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb)
                & aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),Nb) ) )
            & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),Y)),Nb)
            <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb)
                <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),Nb) ) )
            & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)),Nb)
            <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb) ) ) ) ) ).

% word_ops_nth_size
tff(fact_5478_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_5479_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_5480_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_5481_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),
            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)),Xd),Y)),
              $ite(aa(word(A),$o,ord_less(word(A),Xd),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),Xd),Y) ) ).

% bit_twiddle_max
tff(fact_5482_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),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),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),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_5483_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),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_5484_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_5485_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_5486_word__ops__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),zero_zero(nat))
              | aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),zero_zero(nat))
              & aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),Y)),zero_zero(nat))
          <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),zero_zero(nat))
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)),zero_zero(nat))
          <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),zero_zero(nat)) ) ) ) ).

% word_ops_lsb
tff(fact_5487_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_5488_aux,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A2: A,As2: 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_kt(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),P),A2),As2),C2),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),As2)))) = 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_ju(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),P),As2),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),As2)))) ).

% aux
tff(fact_5489_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y222: list(A)] :
      ( ( aa(list(A),list(A),cons(A,X21),X222) = aa(list(A),list(A),cons(A,Y21),Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_5490_length__nth__simps_I4_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% length_nth_simps(4)
tff(fact_5491_nth__Cons__Suc,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% nth_Cons_Suc
tff(fact_5492_length__nth__simps_I3_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),zero_zero(nat)) = Xd ).

% length_nth_simps(3)
tff(fact_5493_nth__Cons__0,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),zero_zero(nat)) = Xd ).

% nth_Cons_0
tff(fact_5494_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222)) = insert(A,X21,aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_5495_nth__Cons__numeral,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),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_5496_nth__Cons__pos,axiom,
    ! [A: $tType,Nb: nat,Xd: A,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_5497_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(A),list(A),cons(A,Y),Ys) )
    <=> ? [Z5: B,Zs: list(B)] :
          ( ( Xs = aa(list(B),list(B),cons(B,Z5),Zs) )
          & ( aa(B,A,F2,Z5) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_5498_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Xd: A,Xs: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),cons(A,Xd),Xs) = aa(list(B),list(A),map(B,A,F2),Ys) )
    <=> ? [Z5: B,Zs: list(B)] :
          ( ( Ys = aa(list(B),list(B),cons(B,Z5),Zs) )
          & ( Xd = aa(B,A,F2,Z5) )
          & ( Xs = aa(list(B),list(A),map(B,A,F2),Zs) ) ) ) ).

% Cons_eq_map_conv
tff(fact_5499_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(A),list(A),cons(A,Y),Ys) )
     => ? [Z2: B,Zs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),cons(B,Z2),Zs2) )
          & ( aa(B,A,F2,Z2) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_5500_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,Xd: A,Xs: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),cons(A,Xd),Xs) = aa(list(B),list(A),map(B,A,F2),Ys) )
     => ? [Z2: B,Zs2: list(B)] :
          ( ( Ys = aa(list(B),list(B),cons(B,Z2),Zs2) )
          & ( Xd = aa(B,A,F2,Z2) )
          & ( Xs = aa(list(B),list(A),map(B,A,F2),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_5501_list_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X21: B,X222: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),cons(B,X21),X222)) = aa(list(A),list(A),cons(A,aa(B,A,F2,X21)),aa(list(B),list(A),map(B,A,F2),X222)) ).

% list.simps(9)
tff(fact_5502_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X222))
     => member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ) ).

% list.set_intros(2)
tff(fact_5503_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : member(A,X21,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ).

% list.set_intros(1)
tff(fact_5504_list_Oset__cases,axiom,
    ! [A: $tType,E: A,A2: list(A)] :
      ( member(A,E,aa(list(A),set(A),set2(A),A2))
     => ( ! [Z23: list(A)] : A2 != aa(list(A),list(A),cons(A,E),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = aa(list(A),list(A),cons(A,Z12),Z23) )
             => ~ member(A,E,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_5505_set__ConsD,axiom,
    ! [A: $tType,Y: A,Xd: A,Xs: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xd),Xs)))
     => ( ( Y = Xd )
        | member(A,Y,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_ConsD
tff(fact_5506_not__Cons__self2,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] : aa(list(A),list(A),cons(A,Xd),Xs) != Xs ).

% not_Cons_self2
tff(fact_5507_length__nth__simps_I2_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,Xd),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_nth_simps(2)
tff(fact_5508_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) )
    <=> ? [Y5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y5),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_5509_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) )
    <=> ? [Y5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y5),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_5510_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xd: A] : aa(set(A),$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),cons(A,Xd),Xs))) ).

% set_subset_Cons
tff(fact_5511_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Xd: A] :
      ( aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))
     => ( Xs != aa(list(A),list(A),cons(A,Xd),Ys) ) ) ).

% impossible_Cons
tff(fact_5512_list__update__code_I3_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),cons(A,Xd),Xs),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),cons(A,Xd),list_update(A,Xs,I,Y)) ).

% list_update_code(3)
tff(fact_5513_list__update__code_I2_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),cons(A,Xd),Xs),zero_zero(nat),Y) = aa(list(A),list(A),cons(A,Y),Xs) ).

% list_update_code(2)
tff(fact_5514_replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xd: A] : replicate(A,aa(nat,nat,suc,Nb),Xd) = aa(list(A),list(A),cons(A,Xd),replicate(A,Nb,Xd)) ).

% replicate_Suc
tff(fact_5515_list__assn_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),A2: A,As2: list(A),C2: B,Cs: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),cons(A,A2),As2)),aa(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,As2),Cs)) ).

% list_assn.simps(2)
tff(fact_5516_list__assn__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),A2: A,As2: list(A),C2: B,Cs: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),cons(A,A2),As2)),aa(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,As2),Cs)) ).

% list_assn_simps(2)
tff(fact_5517_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,X5),Ys3) )
          & aa(nat,$o,ord_less_eq(nat,Nb),aa(list(A),nat,size_size(list(A)),Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_5518_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_5519_nth__Cons_H,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),Nb) = $ite(Nb = zero_zero(nat),Xd,aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ).

% nth_Cons'
tff(fact_5520_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xd: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xd != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) = Y )
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5521_nth__equal__first__eq,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Nb: nat] :
      ( ~ member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$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),cons(A,Xd),Xs)),Nb) = Xd )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5522_Cons__replicate__eq,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( aa(list(A),list(A),cons(A,Xd),Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xd = Y )
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
        & ( Xs = replicate(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xd) ) ) ) ).

% Cons_replicate_eq
tff(fact_5523_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,Xd: A,Xs: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),cons(A,Xd),Xs)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),End) ),
        aa(list(A),list(A),cons(A,Xd),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_5524_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_5525_lsb__odd,axiom,
    ! [A: $tType] :
      ( least_6119777620449941438nt_lsb(A)
     => ! [X4: A] :
          ( aa(A,$o,least_8051144512741203767sb_lsb(A),X4)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),X4) ) ) ).

% lsb_odd
tff(fact_5526_length__Cons,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,Xd),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_5527_int__lsb__numeral_I1_J,axiom,
    ~ aa(int,$o,least_8051144512741203767sb_lsb(int),zero_zero(int)) ).

% int_lsb_numeral(1)
tff(fact_5528_int__lsb__numeral_I2_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),one_one(int)) ).

% int_lsb_numeral(2)
tff(fact_5529_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_5530_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_5531_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_5532_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_5533_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_5534_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_5535_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_5536_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_5537_lsb__integer__code,axiom,
    ! [Xd: code_integer] :
      ( aa(code_integer,$o,least_8051144512741203767sb_lsb(code_integer),Xd)
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(code_integer,Xd),zero_zero(nat)) ) ).

% lsb_integer_code
tff(fact_5538_lsb__int__def,axiom,
    ! [I: int] :
      ( aa(int,$o,least_8051144512741203767sb_lsb(int),I)
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,I),zero_zero(nat)) ) ).

% lsb_int_def
tff(fact_5539_word__lsb__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),zero_zero(nat)) ) ) ).

% word_lsb_alt
tff(fact_5540_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_5541_bin__last__conv__lsb,axiom,
    aTP_Lamp_ku(int,$o) = least_8051144512741203767sb_lsb(int) ).

% bin_last_conv_lsb
tff(fact_5542_lsb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X4: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),X4)
        <=> ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X4) ) ) ).

% lsb_word_eq
tff(fact_5543_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_5544_word__lsb__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> ( 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_5545_word__lsb__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> ( 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_5546_hash__code__prod__simps,axiom,
    ! [A: $tType,B: $tType,H_a: fun(A,uint32),H_b: fun(B,uint32),Xd: A,Xaa: B] : hash_hash_code_prod(A,B,H_a,H_b,aa(B,product_prod(A,B),product_Pair(A,B,Xd),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_a,Xd)),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_5547_vebt__buildup_Opelims,axiom,
    ! [Xd: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xd) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,Xd)
       => ( ( ( Xd = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5548_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X7: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xs)),X7)
     => ( aa(set(A),$o,finite_finite2(A),X7)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,count_list(A,Xs)),X7) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_5549_count__notin,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( ~ member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xd) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_5550_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xd: A] : aa(nat,$o,ord_less_eq(nat,aa(A,nat,count_list(A,Xs),Xd)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% count_le_length
tff(fact_5551_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Y: A] :
      aa(A,nat,count_list(A,aa(list(A),list(A),cons(A,Xd),Xs)),Y) = $ite(Xd = 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_5552_hash__code__option__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),Xd: A] : hash_h1887023736457453652option(A,H_a,aa(A,option(A),some(A),Xd)) = 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_a,Xd)),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_5553_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,
    ! [Xd: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xd) = Y )
     => ( accp(nat,vEBT_V5144397997797733112_d_rel,Xd)
       => ( ( ( Xd = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ accp(nat,vEBT_V5144397997797733112_d_rel,zero_zero(nat)) ) )
         => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_5554_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,
    ! [Xd: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xd) = Y )
     => ( accp(nat,vEBT_V1247956027447740395_p_rel,Xd)
       => ( ( ( Xd = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_V1247956027447740395_p_rel,zero_zero(nat)) ) )
         => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_5555_hash__code__option__simps_I1_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32)] : hash_h1887023736457453652option(A,H_a,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_5556_hash__code__list__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),Xd: A,Xaa: list(A)] : hash_hash_code_list(A,H_a,aa(list(A),list(A),cons(A,Xd),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_a,Xd)),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_a,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_5557_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xd) = Y )
     => ( accp(nat,vEBT_V2957053500504383685pi_rel,Xd)
       => ( ( ( Xd = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ accp(nat,vEBT_V2957053500504383685pi_rel,zero_zero(nat)) ) )
         => ( ( ( Xd = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xd = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) )
                   => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_5558_dup__1,axiom,
    code_dup(one_one(code_integer)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% dup_1
tff(fact_5559_Code__Numeral_Odup__code_I1_J,axiom,
    code_dup(zero_zero(code_integer)) = zero_zero(code_integer) ).

% Code_Numeral.dup_code(1)
tff(fact_5560_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo: option(product_prod(nat,nat)),Deg: nat,Tree_list: list(vEBT_VEBT),Summary: vEBT_VEBT,Mmoi: option(product_prod(nat,nat)),Degi: nat,Tree_array: array(vEBT_VEBTi),Summaryi: vEBT_VEBTi] :
      aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(Mmo,Deg,Tree_list,Summary)),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,Summary),Summaryi))),
        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_kv(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list),Tree_array))) ).

% vebt_assn_raw.simps(2)
tff(fact_5561_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_5562_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_kw(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_5563_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_kx(assn,fun(A,assn),C2)) = C2 ).

% ex_assn_const
tff(fact_5564_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_dm(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ).

% norm_assertion_simps(16)
tff(fact_5565_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_ky(assn,fun(fun(A,assn),fun(A,assn)),R),Q)) ).

% norm_assertion_simps(17)
tff(fact_5566_triv__exI,axiom,
    ! [A: $tType,Q: fun(A,assn),Xd: A] : entails(aa(A,assn,Q,Xd),ex_assn(A,Q)) ).

% triv_exI
tff(fact_5567_mod__ex__dist,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,P)),H)
    <=> ? [X5: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P,X5)),H) ) ).

% mod_ex_dist
tff(fact_5568_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( ( archimedean_frac(A,Xd) = zero_zero(A) )
        <=> member(A,Xd,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_5569_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,Y: A] :
          ( ( member(A,Xd,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),Xd),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xd)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_5570_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),archimedean_frac(A,Xd))
        <=> ~ member(A,Xd,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_5571_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_kz(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M),K),V)) = insert(A,V,ran(B,A,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_5572_mod__h__bot__iff_I8_J,axiom,
    ! [A: $tType,R: fun(A,assn),H: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H),bot_bot(set(nat))))
    <=> ? [X5: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,R,X5)),aa(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_5573_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_5574_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_5575_ex__distrib__star,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_dm(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_5576_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_5577_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_5578_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_5579_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_5580_ex__one__point__gen,axiom,
    ! [A: $tType,P: fun(A,assn),V: A] :
      ( ! [H3: product_prod(heap_ext(product_unit),set(nat)),X: A] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P,X)),H3)
         => ( X = V ) )
     => ( ex_assn(A,P) = aa(A,assn,P,V) ) ) ).

% ex_one_point_gen
tff(fact_5581_mod__exI,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( ? [X4: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P,X4)),H)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,P)),H) ) ).

% mod_exI
tff(fact_5582_mod__exE,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,P)),H)
     => ~ ! [X: A] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P,X)),H) ) ).

% mod_exE
tff(fact_5583_enorm__exI_H,axiom,
    ! [A: $tType,Z6: fun(A,$o),P: assn,Q: fun(A,assn)] :
      ( ! [X: A] :
          ( aa(A,$o,Z6,X)
         => entails(P,aa(A,assn,Q,X)) )
     => ( ? [X_12: A] : aa(A,$o,Z6,X_12)
       => entails(P,ex_assn(A,Q)) ) ) ).

% enorm_exI'
tff(fact_5584_ent__ex__preI,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] :
      ( ! [X: A] : entails(aa(A,assn,P,X),Q)
     => entails(ex_assn(A,P),Q) ) ).

% ent_ex_preI
tff(fact_5585_ent__ex__postI,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xd: A] :
      ( entails(P,aa(A,assn,Q,Xd))
     => entails(P,ex_assn(A,Q)) ) ).

% ent_ex_postI
tff(fact_5586_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_5587_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_5588_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,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_5589_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_5590_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] :
          ( member(A,Xd,ring_1_Ints(A))
         => ( ( Xd != zero_zero(A) )
           => aa(A,$o,ord_less_eq(A,one_one(A)),abs_abs(A,Xd)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5591_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A] :
          ( member(A,Xd,ring_1_Ints(A))
         => ( aa(A,$o,ord_less(A,abs_abs(A,Xd)),one_one(A))
           => ( Xd = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5592_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A] :
          ( member(A,Xd,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xd = Y )
            <=> aa(A,$o,ord_less(A,abs_abs(A,aa(A,A,minus_minus(A,Xd),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5593_sin__times__pi__eq__0,axiom,
    ! [Xd: real] :
      ( ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xd),pi)) = zero_zero(real) )
    <=> member(real,Xd,ring_1_Ints(real)) ) ).

% sin_times_pi_eq_0
tff(fact_5594_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xd)) = $ite(member(A,Xd,ring_1_Ints(A)),zero_zero(A),aa(A,A,minus_minus(A,one_one(A)),archimedean_frac(A,Xd))) ) ).

% frac_neg
tff(fact_5595_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,ord_less_eq(B,aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,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_5596_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xd: A,A2: A] :
          ( ( archimedean_frac(A,Xd) = A2 )
        <=> ( member(A,aa(A,A,minus_minus(A,Xd),A2),ring_1_Ints(A))
            & aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
            & aa(A,$o,ord_less(A,A2),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_5597_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,ord_less_eq(B,aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),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_5598_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_5599_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_5600_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_la(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_5601_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) )
         => ( heap_Time_return(nat,aa(list(B),nat,size_size(list(B)),Tree_is)) = array_len(A,Tree_array) ) ) ) ).

% length_corresp
tff(fact_5602_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel2,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel2,vEBT_Leaf((A5),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( 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,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList2)),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel2,vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_5603_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_5604_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_5605_ent__true,axiom,
    ! [P: assn] : entails(P,top_top(assn)) ).

% ent_true
tff(fact_5606_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_5607_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn))) ) ).

% ent_true_drop(2)
tff(fact_5608_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn)))
     => 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_5609_ent__refl__true,axiom,
    ! [A3: assn] : entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),top_top(assn))) ).

% ent_refl_true
tff(fact_5610_ent__star__mono__true,axiom,
    ! [A3: assn,A6: assn,B4: assn,B10: assn] :
      ( entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A6),top_top(assn)))
     => ( entails(B4,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B10),top_top(assn)))
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B4)),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),A6),B10)),top_top(assn))) ) ) ).

% ent_star_mono_true
tff(fact_5611_mod__star__trueI,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H) ) ).

% mod_star_trueI
tff(fact_5612_mod__star__trueE,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H)
     => ~ ! [H5: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H5) ) ).

% mod_star_trueE
tff(fact_5613_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(top_top(assn)),aa(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_5614_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel,vEBT_Leaf((A5),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( 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,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList2)),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel,vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_5615_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = one_one(real) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2,vEBT_Leaf((A5),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( 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,Summary2))),aa(real,real,foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList2)),zero_zero(real))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2,vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_5616_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,vEBT_Leaf((A5),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( 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,Summary2))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList2)),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_5617_vebt__maxt_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_maxt_rel,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      aa(nat,option(nat),some(nat),one_one(nat)),
                      $ite((A5),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma2) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_5618_vebt__mint_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_mint_rel,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = $ite(
                      (A5),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi) )
                   => ~ 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),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_5619_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A5),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_5620_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(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)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Leaf((A5),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_5621_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,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,Xd)
       => ( ( ( Xd = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one(nat) )
             => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xd = vEBT_Leaf($true,(Uv2)) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xd = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_5622_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xd)
      <=> (Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xd)
       => ( ( ( Xd = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xd = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xd = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                      ( ( Xd = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) )
                     => ( ~ (Y)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_5623_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xd)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xd)
       => ( ( ( Xd = vEBT_Leaf($false,$false) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_5624_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( bits_b8758750999018896077nteger(I)
    <=> ( 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_5625_sofl__test,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,Xd)),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)),Xd),Y)) )
        <=> ( bit_se4197421643247451524op_bit(word(A),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),Xd)),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)),Xd),Y)),Xd)),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)),Xd),Y)),Y))) = zero_zero(word(A)) ) ) ) ).

% sofl_test
tff(fact_5626_bin__rest__integer_Oabs__eq,axiom,
    ! [Xd: int] : bits_b2549910563261871055nteger(code_integer_of_int(Xd)) = code_integer_of_int(divide_divide(int,Xd,aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% bin_rest_integer.abs_eq
tff(fact_5627_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se4197421643247451524op_bit(A,Nb,zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_5628_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,M,bit_se4197421643247451524op_bit(A,Nb,A2)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb),A2) ) ).

% drop_bit_drop_bit
tff(fact_5629_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Nb,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,Nb,A2)),bit_se4197421643247451524op_bit(A,Nb,B2)) ) ).

% drop_bit_and
tff(fact_5630_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Nb,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se4197421643247451524op_bit(A,Nb,A2)),bit_se4197421643247451524op_bit(A,Nb,B2)) ) ).

% drop_bit_or
tff(fact_5631_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Nb,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),bit_se4197421643247451524op_bit(A,Nb,A2)),bit_se4197421643247451524op_bit(A,Nb,B2)) ) ).

% drop_bit_xor
tff(fact_5632_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,B2: $o] :
          bit_se4197421643247451524op_bit(A,Nb,aa($o,A,zero_neq_one_of_bool(A),(B2))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Nb = zero_zero(nat) )
            & (B2) )) ) ).

% drop_bit_of_bool
tff(fact_5633_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),bit0(K))) = bit_se4197421643247451524op_bit(A,Nb,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_5634_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = bit_se4197421643247451524op_bit(A,Nb,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_5635_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se4197421643247451524op_bit(A,Nb,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_5636_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),bit0(K))) = bit_se4197421643247451524op_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_5637_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = bit_se4197421643247451524op_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_5638_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,M: nat] : bit_se4197421643247451524op_bit(A,Nb,aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,Nb,M)) ) ).

% drop_bit_of_nat
tff(fact_5639_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,M,Nb)) = bit_se4197421643247451524op_bit(A,M,aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_drop_bit
tff(fact_5640_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = A2 )
        <=> ( bit_se4197421643247451524op_bit(A,Nb,A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_5641_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4197421643247451524op_bit(A,Nb,A2)) = 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_5642_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: 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)),bit_se4197421643247451524op_bit(A,M,A2)) ) ).

% drop_bit_take_bit
tff(fact_5643_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_5644_bit__word__iff__drop__bit__and,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat] :
          ( aa(nat,$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)),bit_se4197421643247451524op_bit(word(A),Nb,A2)),one_one(word(A))) = one_one(word(A)) ) ) ) ).

% bit_word_iff_drop_bit_and
tff(fact_5645_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,Nb,A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_5646_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_5647_bitAND__integer__unfold,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),Xd),Y) = $ite(
        Xd = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(
          Xd = 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(Xd)),bits_b2549910563261871055nteger(Y)),
            ( bits_b8758750999018896077nteger(Xd)
            & bits_b8758750999018896077nteger(Y) )) ) ) ).

% bitAND_integer_unfold
tff(fact_5648_bitOR__integer__unfold,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),Xd),Y) = $ite(
        Xd = zero_zero(code_integer),
        Y,
        $ite(
          Xd = 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(Xd)),bits_b2549910563261871055nteger(Y)),
            ( bits_b8758750999018896077nteger(Xd)
            | bits_b8758750999018896077nteger(Y) )) ) ) ).

% bitOR_integer_unfold
tff(fact_5649_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) = divide_divide(A,bit_se4197421643247451524op_bit(A,Nb,A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_5650_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( bit_se4197421643247451524op_bit(A,Nb,A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_5651_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb),A2) = bit_se4197421643247451524op_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_5652_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,Nb,A2) = 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_5653_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))),bit_se4197421643247451524op_bit(A,Nb,A2))
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_5654_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se4197421643247451524op_bit(A,Nb,A2)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_5655_bitXOR__integer__unfold,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),Xd),Y) = $ite(
        Xd = zero_zero(code_integer),
        Y,
        $ite(
          Xd = 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(Xd)),bits_b2549910563261871055nteger(Y)),
            ~ bits_b8758750999018896077nteger(Xd) = bits_b8758750999018896077nteger(Y)) ) ) ).

% bitXOR_integer_unfold
tff(fact_5656_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          bit_se4197421643247451524op_bit(A,Nb,A2) = $ite(Nb = zero_zero(nat),A2,bit_se4197421643247451524op_bit(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_5657_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = divide_divide(code_integer,I,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) ).

% bin_rest_integer_code
tff(fact_5658_bin__last__integer_Oabs__eq,axiom,
    ! [Xd: int] :
      ( bits_b8758750999018896077nteger(code_integer_of_int(Xd))
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xd) ) ).

% bin_last_integer.abs_eq
tff(fact_5659_div__half__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),product_Pair(word(A),word(A),divide_divide(word(A),Xd,Y)),modulo_modulo(word(A),Xd,Y)) = $let(
                q: word(A),
                q:= bit_se4730199178511100633sh_bit(word(A),one_one(nat),divide_divide(word(A),bit_se4197421643247451524op_bit(word(A),one_one(nat),Xd),Y)),
                $let(
                  r: word(A),
                  r:= aa(word(A),word(A),minus_minus(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                  $ite(aa(word(A),$o,ord_less_eq(word(A),Y),r),aa(word(A),product_prod(word(A),word(A)),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)),product_Pair(word(A),word(A),q),r)) ) ) ) ) ) ).

% div_half_word
tff(fact_5660_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,ord_less(int,J),I),Js,upto_aux(I,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),cons(int,J),Js))) ).

% upto_aux_rec
tff(fact_5661_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,Nb),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_concat_bit(Nb,divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),L))) ).

% concat_bit_Suc
tff(fact_5662_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se4730199178511100633sh_bit(A,Nb,zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_5663_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit(A,Nb,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_5664_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se4197421643247451524op_bit(int,Nb,K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_5665_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se4197421643247451524op_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_5666_drop__bit__minus__one,axiom,
    ! [Nb: nat] : 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_5667_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,M,bit_se4730199178511100633sh_bit(A,Nb,A2)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb),A2) ) ).

% push_bit_push_bit
tff(fact_5668_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4730199178511100633sh_bit(A,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_and
tff(fact_5669_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se4730199178511100633sh_bit(A,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_or
tff(fact_5670_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),bit_se4730199178511100633sh_bit(A,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_xor
tff(fact_5671_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_5672_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_concat_bit(Nb,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) ).

% concat_bit_of_zero_2
tff(fact_5673_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_concat_bit(Nb,K),L))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_5674_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_concat_bit(Nb,K),L)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_5675_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),K)) = bit_se4730199178511100633sh_bit(A,Nb,aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_Suc_numeral
tff(fact_5676_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = 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_5677_drop__bit__of__Suc__0,axiom,
    ! [Nb: 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_5678_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = 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_5679_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : 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_5680_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),K)) = bit_se4730199178511100633sh_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_numeral
tff(fact_5681_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : 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)))) = 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_5682_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : 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)))) = 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_5683_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),A2) = 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_5684_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : 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_5685_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))),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_5686_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = 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_5687_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : 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)))) = 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_5688_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : 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_5689_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : bit_se4197421643247451524op_bit(nat,Nb,nat2(K)) = nat2(bit_se4197421643247451524op_bit(int,Nb,K)) ).

% drop_bit_nat_eq
tff(fact_5690_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : bit_se4197421643247451524op_bit(int,zero_zero(nat),I) = I ).

% drop_bit_int_code(1)
tff(fact_5691_drop__bit__int__code_I2_J,axiom,
    ! [Nb: nat] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb),zero_zero(int)) = zero_zero(int) ).

% drop_bit_int_code(2)
tff(fact_5692_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: 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)),bit_se4730199178511100633sh_bit(A,M,A2)) ) ).

% push_bit_take_bit
tff(fact_5693_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,M: nat,L: int,R2: int] : aa(int,int,bit_concat_bit(Nb,K),aa(int,int,bit_concat_bit(M,L),R2)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb),aa(int,int,bit_concat_bit(Nb,K),L)),R2) ).

% concat_bit_assoc
tff(fact_5694_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: 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),bit_se4730199178511100633sh_bit(A,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_add
tff(fact_5695_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),bit_se4730199178511100633sh_bit(A,Nb,A2)) ) ).

% push_bit_minus
tff(fact_5696_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : bit_se4730199178511100633sh_bit(A,Nb,aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),bit_se4730199178511100633sh_bit(int,Nb,K)) ) ).

% push_bit_of_int
tff(fact_5697_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B2: int] : bit_concat_bit(Nb,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),B2)) = bit_concat_bit(Nb,B2) ).

% concat_bit_take_bit_eq
tff(fact_5698_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R2: int,S: int] :
      ( ( aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,bit_concat_bit(Nb,R2),S) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),R2) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_5699_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat] : bit_se4730199178511100633sh_bit(A,Nb,aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,Nb,M)) ) ).

% push_bit_of_nat
tff(fact_5700_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,M,Nb)) = bit_se4730199178511100633sh_bit(A,M,aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_push_bit
tff(fact_5701_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4730199178511100633sh_bit(A,Nb,A2)) = 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_5702_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : divide_divide(A,A2,bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) = bit_se4197421643247451524op_bit(A,Nb,A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_5703_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,Nb,bit_se4197421643247451524op_bit(A,Nb,A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = A2 ) ).

% bits_ident
tff(fact_5704_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Nb),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) ) ).

% set_bit_eq_or
tff(fact_5705_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),bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_5706_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: 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),bit_se4730199178511100633sh_bit(A,Nb,A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_5707_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_5708_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] : 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_5709_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,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),bit_se4730199178511100633sh_bit(A,Nb,one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_5710_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(M,K),L)),Nb)
    <=> ( ( aa(nat,$o,ord_less(nat,Nb),M)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) )
        | ( aa(nat,$o,ord_less_eq(nat,M),Nb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5711_shiftr__integer__conv__div__pow2,axiom,
    ! [Nb: nat,Xd: code_integer] : bit_se4197421643247451524op_bit(code_integer,Nb,Xd) = divide_divide(code_integer,Xd,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_5712_bin__rest__code,axiom,
    ! [I: int] : divide_divide(int,I,aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se4197421643247451524op_bit(int,one_one(nat),I) ).

% bin_rest_code
tff(fact_5713_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se4197421643247451524op_bit(int,Nb,K) = 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_5714_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,bit_concat_bit(Nb,K),aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_5715_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: 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_5716_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 != bit_se4730199178511100633sh_bit(A,Nb,B3) ) ) ).

% exp_dvdE
tff(fact_5717_drop__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : bit_se4197421643247451524op_bit(nat,Nb,M) = 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_5718_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,M: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,bit_se2584673776208193580ke_bit(A,M),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_5719_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_lb(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_5720_word__and__mask__or__conv__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),Index: nat] :
          ( aa(nat,$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))),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_5721_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,bit_se5641148757651400278ts_bit(A,l),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),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_5722_word__type__copy__bits__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( cl_HOL_Oequal(A)
        & linorder(A)
        & comm_ring_1(A)
        & semiring_modulo(A)
        & type_len(B) )
     => ! [Word_of: fun(A,word(B)),Signed_drop_bit: fun(nat,fun(A,A))] :
          ( ! [P5: A] : bit_se5641148757651400278ts_bit(A,P5) = bit_se5641148757651400278ts_bit(word(B),aa(A,word(B),Word_of,P5))
         => ( ! [P5: A] : aa(A,word(B),Word_of,aa(A,A,bit_ri4277139882892585799ns_not(A),P5)) = aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),aa(A,word(B),Word_of,P5))
           => ( ! [P5: A,Q4: A] : aa(A,word(B),Word_of,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),P5),Q4)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),aa(A,word(B),Word_of,P5)),aa(A,word(B),Word_of,Q4))
             => ( ! [P5: A,Q4: A] : aa(A,word(B),Word_of,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),P5),Q4)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se1065995026697491101ons_or(word(B)),aa(A,word(B),Word_of,P5)),aa(A,word(B),Word_of,Q4))
               => ( ! [P5: A,Q4: A] : aa(A,word(B),Word_of,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),P5),Q4)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344971392196577ns_xor(word(B)),aa(A,word(B),Word_of,P5)),aa(A,word(B),Word_of,Q4))
                 => ( ! [N: nat] : aa(A,word(B),Word_of,bit_se2239418461657761734s_mask(A,N)) = bit_se2239418461657761734s_mask(word(B),N)
                   => ( ! [N: nat,P5: A] : aa(A,word(B),Word_of,bit_se4730199178511100633sh_bit(A,N,P5)) = bit_se4730199178511100633sh_bit(word(B),N,aa(A,word(B),Word_of,P5))
                     => ( ! [N: nat,P5: A] : aa(A,word(B),Word_of,bit_se4197421643247451524op_bit(A,N,P5)) = bit_se4197421643247451524op_bit(word(B),N,aa(A,word(B),Word_of,P5))
                       => ( ! [N: nat,P5: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),Signed_drop_bit,N),P5)) = signed_drop_bit(B,N,aa(A,word(B),Word_of,P5))
                         => ( ! [N: nat,P5: A] : aa(A,word(B),Word_of,aa(A,A,bit_se2584673776208193580ke_bit(A,N),P5)) = aa(word(B),word(B),bit_se2584673776208193580ke_bit(word(B),N),aa(A,word(B),Word_of,P5))
                           => ( ! [N: nat,P5: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),P5)) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se5668285175392031749et_bit(word(B)),N),aa(A,word(B),Word_of,P5))
                             => ( ! [N: nat,P5: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),P5)) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se2638667681897837118et_bit(word(B)),N),aa(A,word(B),Word_of,P5))
                               => ( ! [N: nat,P5: A] : aa(A,word(B),Word_of,bit_se8732182000553998342ip_bit(A,N,P5)) = bit_se8732182000553998342ip_bit(word(B),N,aa(A,word(B),Word_of,P5))
                                 => word_T1405001541757447525axioms(A,B,Word_of,Signed_drop_bit) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% word_type_copy_bits_axioms.intro
tff(fact_5723_word__type__copy__bits__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( cl_HOL_Oequal(A)
        & linorder(A)
        & comm_ring_1(A)
        & semiring_modulo(A)
        & type_len(B) )
     => ! [Word_of: fun(A,word(B)),Signed_drop_bit: fun(nat,fun(A,A))] :
          ( word_T1405001541757447525axioms(A,B,Word_of,Signed_drop_bit)
        <=> ( ! [P6: A] : bit_se5641148757651400278ts_bit(A,P6) = bit_se5641148757651400278ts_bit(word(B),aa(A,word(B),Word_of,P6))
            & ! [P6: A] : aa(A,word(B),Word_of,aa(A,A,bit_ri4277139882892585799ns_not(A),P6)) = aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),aa(A,word(B),Word_of,P6))
            & ! [P6: A,Q5: A] : aa(A,word(B),Word_of,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),P6),Q5)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),aa(A,word(B),Word_of,P6)),aa(A,word(B),Word_of,Q5))
            & ! [P6: A,Q5: A] : aa(A,word(B),Word_of,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),P6),Q5)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se1065995026697491101ons_or(word(B)),aa(A,word(B),Word_of,P6)),aa(A,word(B),Word_of,Q5))
            & ! [P6: A,Q5: A] : aa(A,word(B),Word_of,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),P6),Q5)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344971392196577ns_xor(word(B)),aa(A,word(B),Word_of,P6)),aa(A,word(B),Word_of,Q5))
            & ! [N6: nat] : aa(A,word(B),Word_of,bit_se2239418461657761734s_mask(A,N6)) = bit_se2239418461657761734s_mask(word(B),N6)
            & ! [N6: nat,P6: A] : aa(A,word(B),Word_of,bit_se4730199178511100633sh_bit(A,N6,P6)) = bit_se4730199178511100633sh_bit(word(B),N6,aa(A,word(B),Word_of,P6))
            & ! [N6: nat,P6: A] : aa(A,word(B),Word_of,bit_se4197421643247451524op_bit(A,N6,P6)) = bit_se4197421643247451524op_bit(word(B),N6,aa(A,word(B),Word_of,P6))
            & ! [N6: nat,P6: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),Signed_drop_bit,N6),P6)) = signed_drop_bit(B,N6,aa(A,word(B),Word_of,P6))
            & ! [N6: nat,P6: A] : aa(A,word(B),Word_of,aa(A,A,bit_se2584673776208193580ke_bit(A,N6),P6)) = aa(word(B),word(B),bit_se2584673776208193580ke_bit(word(B),N6),aa(A,word(B),Word_of,P6))
            & ! [N6: nat,P6: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N6),P6)) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se5668285175392031749et_bit(word(B)),N6),aa(A,word(B),Word_of,P6))
            & ! [N6: nat,P6: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N6),P6)) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se2638667681897837118et_bit(word(B)),N6),aa(A,word(B),Word_of,P6))
            & ! [N6: nat,P6: A] : aa(A,word(B),Word_of,bit_se8732182000553998342ip_bit(A,N6,P6)) = bit_se8732182000553998342ip_bit(word(B),N6,aa(A,word(B),Word_of,P6)) ) ) ) ).

% word_type_copy_bits_axioms_def
tff(fact_5724_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Nb: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F2,aa(nat,nat,suc,Nb),W) = code_T2661198915054445665ts_aux(A,F2,Nb,
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),W)),
              $ite(aa(nat,$o,F2,Nb),one_one(word(A)),zero_zero(word(A))))) ) ).

% set_bits_aux_Suc
tff(fact_5725_signed__drop__bit__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed_drop_bit(A,zero_zero(nat),W) = W ) ).

% signed_drop_bit_0
tff(fact_5726_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_5727_signed__drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : signed_drop_bit(A,Nb,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% signed_drop_bit_of_0
tff(fact_5728_set__bits__aux__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),W: word(A)] : code_T2661198915054445665ts_aux(A,F2,zero_zero(nat),W) = W ) ).

% set_bits_aux_0
tff(fact_5729_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se4730199178511100633sh_bit(int,Nb,K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_5730_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se4730199178511100633sh_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_5731_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_5732_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa(int,int,bit_concat_bit(Nb,zero_zero(int)),L) = bit_se4730199178511100633sh_bit(int,Nb,L) ).

% concat_bit_of_zero_1
tff(fact_5733_push__bit__of__Suc__0,axiom,
    ! [Nb: 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_5734_drop__bit__push__bit__int,axiom,
    ! [M: nat,Nb: nat,K: int] : bit_se4197421643247451524op_bit(int,M,bit_se4730199178511100633sh_bit(int,Nb,K)) = bit_se4197421643247451524op_bit(int,aa(nat,nat,minus_minus(nat,M),Nb),bit_se4730199178511100633sh_bit(int,aa(nat,nat,minus_minus(nat,Nb),M),K)) ).

% drop_bit_push_bit_int
tff(fact_5735_push__bit__int__code_I1_J,axiom,
    ! [I: int] : bit_se4730199178511100633sh_bit(int,zero_zero(nat),I) = I ).

% push_bit_int_code(1)
tff(fact_5736_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : bit_se4730199178511100633sh_bit(nat,Nb,nat2(K)) = nat2(bit_se4730199178511100633sh_bit(int,Nb,K)) ).

% push_bit_nat_eq
tff(fact_5737_set__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),M),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% set_bit_nat_def
tff(fact_5738_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),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% flip_bit_nat_def
tff(fact_5739_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$false) = bit_se4730199178511100633sh_bit(code_integer,one_one(nat),I) ).

% Bit_integer_code(1)
tff(fact_5740_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,bit_se4730199178511100633sh_bit(int,M,K)),Nb)
    <=> ( aa(nat,$o,ord_less_eq(nat,M),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_5741_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,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)),bit_se4730199178511100633sh_bit(int,Nb,L)) ).

% concat_bit_eq
tff(fact_5742_concat__bit__def,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),bit_se4730199178511100633sh_bit(int,Nb,L)) ).

% concat_bit_def
tff(fact_5743_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),bit_se4730199178511100633sh_bit(int,Nb,one_one(int))) ).

% Bit_Operations.set_bit_int_def
tff(fact_5744_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,M,Q2)),Nb)
    <=> ( aa(nat,$o,ord_less_eq(nat,M),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q2),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_5745_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),bit_se4730199178511100633sh_bit(int,Nb,one_one(int))) ).

% flip_bit_int_def
tff(fact_5746_shiftl__integer__conv__mult__pow2,axiom,
    ! [Nb: nat,Xd: code_integer] : bit_se4730199178511100633sh_bit(code_integer,Nb,Xd) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),Xd),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_5747_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,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),bit_se4730199178511100633sh_bit(int,Nb,one_one(int)))) ).

% unset_bit_int_def
tff(fact_5748_push__bit__int__def,axiom,
    ! [Nb: nat,K: 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_5749_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),bit_se4730199178511100633sh_bit(code_integer,one_one(nat),I)),one_one(code_integer)) ).

% Bit_integer_code(2)
tff(fact_5750_push__bit__nat__def,axiom,
    ! [Nb: nat,M: 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_5751_push__bit__minus__one,axiom,
    ! [Nb: nat] : 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_5752_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Nb: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F2,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,F2,n,
                aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),W)),
                  $ite(aa(nat,$o,F2,n),one_one(word(A)),zero_zero(word(A))))) ) ) ) ).

% set_bits_aux_rec
tff(fact_5753_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),product_Pair(word(A),word(B),A2),B2) )
         => ( ! [N9: nat] :
                ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(B),B2),N9)
              <=> ( aa(nat,$o,ord_less(nat,N9),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,bit_se5641148757651400278ts_bit(word(C),C2),N9) ) )
            & ! [M3: nat] :
                ( aa(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,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_5754_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),product_Pair(word(A),word(B),A2),B2) )
         => ! [N9: nat,M3: nat] :
              ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(B),B2),N9)
              <=> ( aa(nat,$o,ord_less(nat,N9),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,bit_se5641148757651400278ts_bit(word(C),C2),N9) ) )
              & ( aa(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,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_5755_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),product_Pair(word(A),word(B),A2),B2) )
        <=> ( ! [N6: nat] :
                ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(C),C2),N6) ) )
            & ! [M4: nat] :
                ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),A2),M4)
              <=> ( aa(nat,$o,ord_less(nat,M4),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,bit_se5641148757651400278ts_bit(word(C),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_5756_set__bit__integer__conv__masks,axiom,
    ! [Xd: code_integer,I: nat,B2: $o] :
      generi7602027413899671122et_bit(code_integer,Xd,I,(B2)) = $ite((B2),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),Xd),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),Xd),aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),bit_se4730199178511100633sh_bit(code_integer,I,one_one(code_integer))))) ).

% set_bit_integer_conv_masks
tff(fact_5757_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))),F2: 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,F2),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,F2),n),
                    aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),bit_se4730199178511100633sh_bit(B,one_one(nat),W)),
                      $ite(aa(nat,$o,F2,n),one_one(B),zero_zero(B)))) ) ) ) ) ) ).

% word_type_copy_misc.set_bits_aux_code
tff(fact_5758_Uint32__code,axiom,
    ! [I: code_integer] :
      aa(code_integer,uint32,uint322,I) = $let(
        i: code_integer,
        i:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),
        $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(code_integer,i),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),uint32_signed(aa(code_integer,code_integer,minus_minus(code_integer,i),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))))),uint32_signed(i)) ) ).

% Uint32_code
tff(fact_5759_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( generic_set_set_bit(A)
     => ! [A2: A,M: nat,B2: $o,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,generi7602027413899671122et_bit(A,A2,M,(B2))),Nb)
        <=> ( $ite(M = Nb,(B2),aa(nat,$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_5760_Generic__set__bit_Oset__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations(A)
        & generic_set_set_bit(A) )
     => ! [A2: A,Nb: nat,B2: $o] :
          generi7602027413899671122et_bit(A,A2,Nb,(B2)) = aa(A,A,
            aa(nat,fun(A,A),
              $ite((B2),bit_se5668285175392031749et_bit(A),bit_se2638667681897837118et_bit(A)),
              Nb),
            A2) ) ).

% Generic_set_bit.set_bit_eq
tff(fact_5761_Uint32__signed__def,axiom,
    ! [I: code_integer] :
      uint32_signed(I) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,I),aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))),I) ),
        aa(code_integer,uint32,aa(fun(code_integer,uint32),fun(code_integer,uint32),undefined(fun(fun(code_integer,uint32),fun(code_integer,uint32))),uint322),I),
        aa(code_integer,uint32,uint322,I) ) ).

% Uint32_signed_def
tff(fact_5762_set__bit__unfold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat,B2: $o] :
          generi7602027413899671122et_bit(word(A),W,Nb,(B2)) = $ite((B2),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se5668285175392031749et_bit(word(A)),Nb),W),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2638667681897837118et_bit(word(A)),Nb),W)) ) ).

% set_bit_unfold
tff(fact_5763_Generic__set__bit_Oset__bit__int__def,axiom,
    ! [I: int,Nb: nat,B2: $o] :
      generi7602027413899671122et_bit(int,I,Nb,(B2)) = aa(int,int,
        aa(nat,fun(int,int),
          $ite((B2),bit_se5668285175392031749et_bit(int),bit_se2638667681897837118et_bit(int)),
          Nb),
        I) ).

% Generic_set_bit.set_bit_int_def
tff(fact_5764_test__bit__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat,Xd: $o] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),generi7602027413899671122et_bit(word(A),W,Nb,(Xd))),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),W))
            & (Xd) ) ) ) ).

% test_bit_set
tff(fact_5765_test__bit__set__gen,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat,Xd: $o,M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),generi7602027413899671122et_bit(word(A),W,Nb,(Xd))),M)
        <=> $ite(
              M = Nb,
              ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),W))
              & (Xd) ),
              aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),M) ) ) ) ).

% test_bit_set_gen
tff(fact_5766_one__bit__pow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : generi7602027413899671122et_bit(word(A),zero_zero(word(A)),Nb,$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_5767_int__set__bit__True__conv__OR,axiom,
    ! [I: int,Nb: nat] : generi7602027413899671122et_bit(int,I,Nb,$true) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),bit_se4730199178511100633sh_bit(int,Nb,one_one(int))) ).

% int_set_bit_True_conv_OR
tff(fact_5768_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,Nb: nat] : 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),bit_se4730199178511100633sh_bit(int,Nb,one_one(int)))) ).

% int_set_bit_False_conv_NAND
tff(fact_5769_int__set__bit__conv__ops,axiom,
    ! [I: int,Nb: nat,B2: $o] :
      generi7602027413899671122et_bit(int,I,Nb,(B2)) = $ite((B2),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),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),bit_se4730199178511100633sh_bit(int,Nb,one_one(int))))) ).

% int_set_bit_conv_ops
tff(fact_5770_mod__h__bot__normalize,axiom,
    ! [A: $tType,H: heap_ext(product_unit),P: assn] :
      ( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),product_Pair(heap_ext(product_unit),set(nat),H),bot_bot(set(nat))))
      <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),aa(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_5771_word__cat__split__alt,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B)
        & type_len(C) )
     => ! [W: word(A),U: word(B),V: word(C)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V)))
         => ( ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),product_Pair(word(B),word(C),U),V) )
           => ( word_cat(B,C,A,U,V) = W ) ) ) ) ).

% word_cat_split_alt
tff(fact_5772_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(A),U: word(B),V: word(C)] :
          ( ( W = word_cat(B,C,A,U,V) )
         => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V))),aa(word(A),nat,size_size(word(A)),W))
           => ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),product_Pair(word(B),word(C),U),V) ) ) ) ) ).

% word_split_cat_alt
tff(fact_5773_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,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,$o,bit_se5641148757651400278ts_bit(word(C),B2),Nb),aa(nat,$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_5774_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [T: word(A),U: word(B),V: word(C)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),T)),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)),product_Pair(word(B),word(C),U),V) = word_split(A,B,C,T) )
           => ( T = word_cat(B,C,A,U,V) ) ) ) ) ).

% word_cat_split_size
tff(fact_5775_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A2: word(A),Nb: nat,C2: word(B),B2: word(C)] :
          ( ( A2 = slice2(B,A,Nb,C2) )
         => ( ( B2 = slice2(B,C,zero_zero(nat),C2) )
           => ( ( Nb = aa(word(C),nat,size_size(word(C)),B2) )
             => ( aa(nat,$o,ord_less_eq(nat,aa(word(B),nat,size_size(word(B)),C2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(C),nat,size_size(word(C)),B2)))
               => ( word_cat(A,C,B,A2,B2) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_5776_mult_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Xd: A,Y: A,A2: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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_5777_add_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [Xd: A,Y: A,A2: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),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_5778_slice__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat] : slice2(B,A,Nb,zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice_0
tff(fact_5779_slice__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [T: word(A)] : slice2(A,A,zero_zero(nat),T) = T ) ).

% slice_id
tff(fact_5780_assn__aci_I11_J,axiom,
    ! [Xd: assn,Y: assn,A2: assn,B2: assn] :
      ( syntax7388354845996824322omatch(assn,assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Xd),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_5781_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_5782_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_5783_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_5784_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_5785_ucast__slice,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = slice2(B,A,zero_zero(nat),W) ) ).

% ucast_slice
tff(fact_5786_slice__cat2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A2: word(B),T: word(A)] : slice2(A,A,zero_zero(nat),word_cat(B,A,A,A2,T)) = T ) ).

% slice_cat2
tff(fact_5787_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,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)))
         => ( slice2(C,A,aa(word(B),nat,size_size(word(B)),B2),word_cat(A,B,C,A2,B2)) = A2 ) ) ) ).

% slice_cat1
tff(fact_5788_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)),product_Pair(word(A),word(B),U),V) )
         => ( ( U = slice2(C,A,aa(word(B),nat,size_size(word(B)),V),W) )
            & ( V = slice2(C,B,zero_zero(nat),W) ) ) ) ) ).

% split_slices
tff(fact_5789_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F3: set(A),I6: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),F3)
         => ( aa(set(A),$o,ord_less_eq(set(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_lc(set(A),fun(fun(A,B),fun(A,$o)),I6),F2))),F3)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I6),insert(A,I,bot_bot(set(A))))) = $ite(member(A,I,I6),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I6)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I6)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_5790_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),Nb) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_5791_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_5792_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P3,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_5793_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bt(set(A),fun(fun(A,B),fun(A,$o)),I6),P3)))
         => ( groups1027152243600224163dd_sum(A,B,P3,insert(A,I,I6)) = $ite(member(A,I,I6),groups1027152243600224163dd_sum(A,B,P3,I6),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,I)),groups1027152243600224163dd_sum(A,B,P3,I6))) ) ) ) ).

% sum.insert'
tff(fact_5794_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I6: set(B)] : groups1027152243600224163dd_sum(B,A,G,collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ld(fun(B,A),fun(set(B),fun(B,$o)),G),I6))) = groups1027152243600224163dd_sum(B,A,G,I6) ) ).

% sum.non_neutral'
tff(fact_5795_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_le(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I6) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I6)),groups1027152243600224163dd_sum(A,B,H,I6)) ) ) ) ).

% sum.distrib_triv'
tff(fact_5796_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S3) = groups1027152243600224163dd_sum(A,B,G,T4) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_5797_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T4) = groups1027152243600224163dd_sum(A,B,G,S3) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_5798_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
         => ( ! [I3: A] :
                ( member(A,I3,aa(set(A),set(A),minus_minus(set(A),T4),S3))
               => ( aa(A,B,H,I3) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S3) = groups1027152243600224163dd_sum(A,B,H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_5799_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),minus_minus(set(A),T4),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T4) = groups1027152243600224163dd_sum(A,B,H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_5800_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bt(set(A),fun(fun(A,B),fun(A,$o)),I6),G)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bt(set(A),fun(fun(A,B),fun(A,$o)),I6),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_le(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I6) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I6)),groups1027152243600224163dd_sum(A,B,H,I6)) ) ) ) ) ).

% sum.distrib'
tff(fact_5801_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I6: set(B)] :
          groups1027152243600224163dd_sum(B,A,P3,I6) = $ite(aa(set(B),$o,finite_finite2(B),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ld(fun(B,A),fun(set(B),fun(B,$o)),P3),I6))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,P3),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ld(fun(B,A),fun(set(B),fun(B,$o)),P3),I6))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_5802_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I6: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_lc(set(A),fun(fun(A,B),fun(A,$o)),I6),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I6),insert(A,I,bot_bot(set(A))))) = $ite(member(A,I,I6),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I6)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I6)) ) ) ) ).

% sum_diff1'
tff(fact_5803_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_lf(list(A),fun(array(A),assn),Xs)) ) ).

% of_list_rule
tff(fact_5804_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,ord_less(int,groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_5805_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xd: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,aa(list(B),list(B),cons(B,Xd),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),groups4207007520872428315er_sum(B,A,F2,A2,Xs))) ) ).

% horner_sum_simps(2)
tff(fact_5806_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,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_lg(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_5807_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_lh(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_5808_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,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_5809_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),A2)),one_one(nat))))),signed7115095781618012415divide(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))) ) ).

% sdiv_word_min
tff(fact_5810_Cauchy__iff2,axiom,
    ! [X7: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X7)
    <=> ! [J2: nat] :
        ? [M9: nat] :
        ! [M4: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M9),M4)
         => ! [N6: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M9),N6)
             => aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,aa(nat,real,X7,M4)),aa(nat,real,X7,N6)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J2)))) ) ) ) ).

% Cauchy_iff2
tff(fact_5811_int__sdiv__simps_I2_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,zero_zero(int)) = zero_zero(int) ).

% int_sdiv_simps(2)
tff(fact_5812_sdiv__int__0__div,axiom,
    ! [Xd: int] : signed7115095781618012415divide(int,zero_zero(int),Xd) = zero_zero(int) ).

% sdiv_int_0_div
tff(fact_5813_sdiv__int__div__0,axiom,
    ! [Xd: int] : signed7115095781618012415divide(int,Xd,zero_zero(int)) = zero_zero(int) ).

% sdiv_int_div_0
tff(fact_5814_int__sdiv__simps_I1_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,one_one(int)) = A2 ).

% int_sdiv_simps(1)
tff(fact_5815_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_5816_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_5817_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)) = divide_divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb)) ).

% sdiv_int_numeral_numeral
tff(fact_5818_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_5819_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_5820_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_5821_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_5822_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Nb: nat,Ns: list(nat),Q2: nat] :
      ( ( aa(list(nat),list(nat),cons(nat,M),aa(list(nat),list(nat),cons(nat,Nb),Ns)) = upt(M,Q2) )
    <=> ( aa(list(nat),list(nat),cons(nat,Nb),Ns) = upt(aa(nat,nat,suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_5823_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_5824_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_li(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_5825_map__add__upt,axiom,
    ! [Nb: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_li(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_5826_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_5827_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xd: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,Xd),Xs) )
    <=> ( aa(nat,$o,ord_less(nat,I),J)
        & ( I = Xd )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_5828_map__decr__upt,axiom,
    ! [M: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_lj(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = upt(M,Nb) ).

% map_decr_upt
tff(fact_5829_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_5830_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_5831_map__replicate__trivial,axiom,
    ! [A: $tType,Xd: A,I: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_lk(A,fun(nat,A),Xd)),upt(zero_zero(nat),I)) = replicate(A,I,Xd) ).

% map_replicate_trivial
tff(fact_5832_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( signed7115095781618012415divide(int,A2,B2) != zero_zero(int) )
     => ( sgn_sgn(int,signed7115095781618012415divide(int,A2,B2)) = sgn_sgn(int,aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_5833_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_5834_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Nb: nat] : aa(list(nat),list(A),map(nat,A,F2),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(list(A),list(A),cons(A,aa(nat,A,F2,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_ll(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),Nb))) ).

% map_upt_Suc
tff(fact_5835_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),L) )
     => ( aa(list(nat),list($o),map(nat,$o,bit_se5641148757651400278ts_bit(int,K)),upt(zero_zero(nat),Nb)) = aa(list(nat),list($o),map(nat,$o,bit_se5641148757651400278ts_bit(int,L)),upt(zero_zero(nat),Nb)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_5836_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_5837_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Nb: nat,M: nat,F2: fun(nat,A)] :
      ( aa(nat,$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,F2),upt(M,Nb))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_5838_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),sgn_sgn(int,K)),sgn_sgn(int,L))),divide_divide(int,abs_abs(int,K),abs_abs(int,L))) ).

% signed_divide_int_def
tff(fact_5839_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,minus_minus(nat,Nb),M) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I3)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F2),upt(M,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_5840_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X7)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E)
           => ? [M8: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M8),M3)
               => ! [N9: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,M8),N9)
                   => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X7,M3)),aa(nat,A,X7,N9)))),E) ) ) ) ) ) ).

% CauchyD
tff(fact_5841_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
             => ? [M10: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M10),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M10),N)
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X7,M2)),aa(nat,A,X7,N)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% CauchyI
tff(fact_5842_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),M4)
                 => ! [N6: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M9),N6)
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X7,M4)),aa(nat,A,X7,N6)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_5843_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$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_5844_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F2: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,Nb,F2),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_lm(nat,fun(fun(nat,A),fun(array(A),assn)),Nb),F2)) ) ).

% make_rule
tff(fact_5845_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,top_top(assn))
     => entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn))) ) ).

% entails_solve_init(1)
tff(fact_5846_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(vEBT_VEBT),nat,size_list(vEBT_VEBT,size_size(vEBT_VEBT)),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_5847_size__list__estimation,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,ord_less(nat,Y),aa(A,nat,F2,Xd))
       => aa(nat,$o,ord_less(nat,Y),aa(list(A),nat,size_list(A,F2),Xs)) ) ) ).

% size_list_estimation
tff(fact_5848_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,ord_less_eq(nat,aa(A,nat,F2,X)),aa(A,nat,G,X)) )
     => aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_list(A,F2),Xs)),aa(list(A),nat,size_list(A,G),Xs)) ) ).

% size_list_pointwise
tff(fact_5849_size__list__estimation_H,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,ord_less_eq(nat,Y),aa(A,nat,F2,Xd))
       => aa(nat,$o,ord_less_eq(nat,Y),aa(list(A),nat,size_list(A,F2),Xs)) ) ) ).

% size_list_estimation'
tff(fact_5850_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F3: assn] :
      ( fI_QUERY(P,Q,F3)
     => entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F3)) ) ).

% frame_inference_init
tff(fact_5851_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F3: assn] :
      ( fI_QUERY(P,Q,F3)
    <=> entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F3)) ) ).

% FI_QUERY_def
tff(fact_5852_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,one_one(assn))
     => entails(P,Q) ) ).

% entails_solve_init(2)
tff(fact_5853_time__array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F2: fun(nat,A),H: heap_ext(product_unit)] : time_time(array(A),array_make(A,Nb,F2),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ) ).

% time_array_make
tff(fact_5854_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xd: fun(A,nat),X21: A,X222: list(A)] : aa(list(A),nat,size_list(A,Xd),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xd,X21)),aa(list(A),nat,size_list(A,Xd),X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_5855_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(vEBT_VEBT),nat,size_list(vEBT_VEBT,vEBT_size_VEBT),X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_5856_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = aa(nat,nat,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_5857_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),abs_abs(int,B2))),one_one(int)),aa(int,int,minus_minus(int,abs_abs(int,B2)),one_one(int)))) ) ).

% smod_int_range
tff(fact_5858_smod__int__0__mod,axiom,
    ! [Xd: int] : signed6721504322012087516modulo(int,zero_zero(int),Xd) = zero_zero(int) ).

% smod_int_0_mod
tff(fact_5859_smod__int__mod__0,axiom,
    ! [Xd: int] : signed6721504322012087516modulo(int,Xd,zero_zero(int)) = Xd ).

% smod_int_mod_0
tff(fact_5860_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)) = modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb)) ).

% smod_int_numeral_numeral
tff(fact_5861_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_5862_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_5863_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(2)
tff(fact_5864_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_eq(int,signed6721504322012087516modulo(int,A2,B2)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_5865_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(6)
tff(fact_5866_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,signed6721504322012087516modulo(int,A2,B2)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_5867_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,B2),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(8)
tff(fact_5868_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A2)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B2)
       => ( signed6721504322012087516modulo(int,A2,B2) = modulo_modulo(int,A2,B2) ) ) ) ).

% smod_mod_positive
tff(fact_5869_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_5870_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(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_5871_smod__int__alt__def,axiom,
    ! [A2: int,B2: int] : signed6721504322012087516modulo(int,A2,B2) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,A2)),modulo_modulo(int,abs_abs(int,A2),abs_abs(int,B2))) ).

% smod_int_alt_def
tff(fact_5872_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_5873_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_5874_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X) = 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_5875_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(T,D)
     => vEBT_invar_vebt(T,D) ) ).

% valid_eq2
tff(fact_5876_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(T,D)
    <=> vEBT_invar_vebt(T,D) ) ).

% valid_eq
tff(fact_5877_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( vEBT_invar_vebt(T,D)
     => vEBT_VEBT_valid(T,D) ) ).

% valid_eq1
tff(fact_5878_map__concat,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),map(B,A,F2),concat(B,Xs)) = concat(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),Xs)) ).

% map_concat
tff(fact_5879_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_5880_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys: list(A),Xs: list(A)] :
      ( member(list(A),aa(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_5881_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D)
    <=> ( D = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_5882_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),Xss)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_ln(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_5883_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_lo(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_5884_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_lp(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_5885_Multiset_Ois__empty__def,axiom,
    ! [A: $tType,A3: multiset(A)] :
      ( is_empty(A,A3)
    <=> ( A3 = zero_zero(multiset(A)) ) ) ).

% Multiset.is_empty_def
tff(fact_5886_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : 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)),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_5887_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Nb)
         => ( bit_se4197421643247451524op_bit(word(A),Nb,W) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_5888_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Nb)
         => ( bit_se4730199178511100633sh_bit(word(A),Nb,W) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_5889_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_5890_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_5891_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_5892_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$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,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_5893_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_5894_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_5895_bit__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),Nb) ) ) ) ).

% bit_numeral_word_iff
tff(fact_5896_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_5897_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_5898_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_5899_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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_5900_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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_5901_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_5902_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,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Nb) ) ) ).

% exp_eq_zero_iff
tff(fact_5903_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_5904_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_5905_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_5906_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_5907_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_5908_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5909_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5910_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_5911_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),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_5912_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$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,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_5913_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),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_5914_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_5915_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_5916_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_5917_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_5918_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$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,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_5919_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : 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)),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_5920_drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : 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)),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_5921_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_5922_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_5923_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_5924_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_5925_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_5926_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_5927_drop__bit__numeral__bit0__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: num] : 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)),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_5928_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5929_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5930_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5931_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5932_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5933_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : 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)),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_5934_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),one_one(word(A))))
          <=> aa(word(A),$o,ord_less(word(A),Xd),aa(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_5935_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_5936_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)),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_5937_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)),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_5938_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : 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)),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_5939_drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : 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)),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_5940_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)),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_5941_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)),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_5942_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_5943_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_5944_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : 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)),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_5945_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : 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)),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_5946_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : 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)),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_5947_nth__slice,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat,W: word(B),M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),slice2(B,A,Nb,W)),M)
        <=> ( aa(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_5948_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_5949_test__bit__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(int,word(A),ring_1_of_int(word(A)),Xd)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,Xd),Nb) ) ) ) ).

% test_bit_wi
tff(fact_5950_bit__word__of__int__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ) ).

% bit_word_of_int_iff
tff(fact_5951_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_5952_ucast__less__ucast__weak,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xd)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less(word(A),Xd),Y) ) ) ) ).

% ucast_less_ucast_weak
tff(fact_5953_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,ord_less(word(A),Xd),Y)
           => aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xd)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono
tff(fact_5954_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xd)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less(word(A),Xd),Y) ) ) ) ).

% ucast_less_ucast
tff(fact_5955_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(B),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),Xd),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
           => aa(word(A),$o,ord_less(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xd)),Y) ) ) ) ).

% less_ucast_ucast_less
tff(fact_5956_ucast__ucast__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A)] :
          ( aa(nat,$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)),Xd)) = Xd ) ) ) ).

% ucast_ucast_id
tff(fact_5957_degenerate__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
         => ( ( Xd = zero_zero(word(A)) )
            | ( Xd = one_one(word(A)) ) ) ) ) ).

% degenerate_word
tff(fact_5958_zero__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% zero_word.rsp
tff(fact_5959_size__0__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = zero_zero(nat) )
         => ( W = V ) ) ) ).

% size_0_same
tff(fact_5960_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_5961_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_5962_uint__word__arith__bintrs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% uint_word_arith_bintrs(7)
tff(fact_5963_test__bit__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_5964_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_5965_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_5966_bit__imp__le__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$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_5967_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),A2),N)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),B2),N) ) )
         => ( A2 = B2 ) ) ) ).

% bit_word_eqI
tff(fact_5968_word__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ( Xd = Y )
        <=> ! [N6: nat] :
              ( aa(nat,$o,ord_less(nat,N6),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),N6)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),N6) ) ) ) ) ).

% word_eq_iff
tff(fact_5969_test__bit__conj__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),M: nat] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),M)
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) )
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),M) ) ) ).

% test_bit_conj_lt
tff(fact_5970_nth__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),Nb)
        <=> ( aa(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_5971_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(B),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(B),A2),Nb) ) ) ) ).

% bit_ucast_iff
tff(fact_5972_bin__nth__uint__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$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_5973_bit__word__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(B),W),Nb) ) ) ) ).

% bit_word_ucast_iff
tff(fact_5974_neg__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)),Nb)
        <=> ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb)
            & aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_test_bit
tff(fact_5975_num__abs__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] : aa(num,word(A),numeral_numeral(word(A)),Xd) = 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),Xd))) ) ).

% num_abs_bintr
tff(fact_5976_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_5977_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_5978_word__of__int__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,ord_less(word(A),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_iff
tff(fact_5979_max__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_5980_bit__uint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(A),W),Nb) ) ) ) ).

% bit_uint_iff
tff(fact_5981_test__bit__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),Nb) ) ) ) ).

% test_bit_bin
tff(fact_5982_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_5983_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B),C2: word(A),D: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( word_cat(A,B,C,A2,B2) = word_cat(A,B,C,C2,D) )
          <=> ( ( A2 = C2 )
              & ( B2 = D ) ) ) ) ) ).

% word_cat_inj
tff(fact_5984_bit__set__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,B2: $o,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(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,bit_se5641148757651400278ts_bit(word(A),W),Nb) ) ) ) ).

% bit_set_bit_word_iff
tff(fact_5985_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_5986_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Nb: nat,W: word(A),M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),code_T2661198915054445665ts_aux(A,F2,Nb,W)),M)
        <=> ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,ord_less(nat,M),Nb),aa(nat,$o,F2,M),aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,M),Nb))) ) ) ) ).

% bit_set_bit_aux
tff(fact_5987_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,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,$o,bit_se5641148757651400278ts_bit(word(C),W),Nb),aa(nat,$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_5988_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_5989_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_5990_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_5991_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_5992_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Nb)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,ring_1_signed(A,int,W)),Nb)
          <=> aa(nat,$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_5993_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_5994_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$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,$o,ord_less_eq(nat,Nb),M)
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_5995_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [B2: word(B),A2: word(A),D: word(B),C2: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2))) = aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),D)),aa(word(A),int,semiring_1_unsigned(A,int),C2))) )
          <=> ( ( B2 = D )
              & ( A2 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_5996_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A)] :
          ( aa(nat,$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)),Xd),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_5997_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_5998_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_5999_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A)] :
        ? [N: nat] :
          ( ( W3 = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
          & aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nchotomy
tff(fact_6000_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ~ ! [N: nat] :
              ( ( Xd = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
             => ~ aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nat_cases
tff(fact_6001_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),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)),Xd) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
            <=> ( Xd = Y ) ) ) ) ) ).

% of_nat_inj
tff(fact_6002_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),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)),Xd) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
             => ( Xd = Y ) ) ) ) ) ).

% word_of_nat_inj
tff(fact_6003_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_6004_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( aa(word(A),$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))),Xd)),aa(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,$o,ord_less(nat,Xd),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,ord_less(nat,Y),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less(nat,Xd),Y) ) ) ) ) ).

% word_power_increasing
tff(fact_6005_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_6006_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_6007_nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$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_6008_uint__idem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : 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_6009_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_6010_of__nat__neq__iff__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( ( modulo_modulo(nat,Xd,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)))) != 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)),Xd) != aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
          <=> ( Xd != Y ) ) ) ) ).

% of_nat_neq_iff_word
tff(fact_6011_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_6012_num__abs__sbintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] : aa(num,word(A),numeral_numeral(word(A)),Xd) = 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),Xd))) ) ).

% num_abs_sbintr
tff(fact_6013_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(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)),Xd)) = Xd ) ) ) ).

% ucast_ucast_len
tff(fact_6014_ucast__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),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)),Xd)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono
tff(fact_6015_horner__sum__uint__exp__Cons__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Ws: list(word(A))] : groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),aa(list(word(A)),list(word(A)),cons(word(A),W),Ws)) = aa(int,int,bit_concat_bit(type_len0_len_of(A,type2(A)),aa(word(A),int,semiring_1_unsigned(A,int),W)),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)) ) ).

% horner_sum_uint_exp_Cons_eq
tff(fact_6016_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_6017_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_6018_nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),W),Nb),aa(nat,$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_6019_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_6020_bit__sint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,ring_1_signed(A,int,W)),Nb)
        <=> ( ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Nb)
              & aa(nat,$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,bit_se5641148757651400278ts_bit(word(A),W),Nb) ) ) ) ).

% bit_sint_iff
tff(fact_6021_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_6022_drop__bit__eq__zero__iff__not__bit__last,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( bit_se4197421643247451524op_bit(word(A),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),W) = zero_zero(word(A)) )
        <=> ~ aa(nat,$o,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_6023_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_6024_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_6025_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_6026_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)))
         => 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_6027_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
       => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = aa(num,int,numeral_numeral(int),bit0(one2)) ) ) ) ).

% uint_2_id
tff(fact_6028_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$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_6029_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),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,ord_less_eq(nat,Nb),M) ) ) ) ) ).

% power_le_mono
tff(fact_6030_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$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,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_6031_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)) = 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_6032_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),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,$o,ord_less(nat,Y),Xd)
           => aa(word(A),$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)),Xd)) ) ) ) ).

% of_nat_mono_maybe
tff(fact_6033_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),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),Xd)
            <=> 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)),Xd)) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_6034_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xd: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd))
        <=> ! [N6: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N6) = Xd )
                & aa(nat,$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_6035_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: nat,A2: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R2) = A2 )
         => ( aa(nat,$o,ord_less(nat,R2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),A2) = R2 ) ) ) ) ).

% of_nat_inverse
tff(fact_6036_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xd: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd))
        <=> ~ ? [N6: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N6) = Xd )
                & aa(nat,$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_6037_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xd: word(A)] :
          ( aa(nat,$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),Xd) = Nb )
          <=> ( Xd = aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_6038_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),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)),Xd)) = Xd ) ) ) ).

% unat_of_nat_len
tff(fact_6039_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) != one_one(nat) )
         => ( aa(word(A),$o,ord_less(word(A),Xd),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)))
           => ( ( Xd = zero_zero(word(A)) )
              | ( Xd = one_one(word(A)) ) ) ) ) ) ).

% x_less_2_0_1'
tff(fact_6040_test__bit__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$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_6041_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,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_6042_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)) )
        <=> ? [Q5: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),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_6043_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A),Y: word(B)] : aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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_6044_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_6045_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)) = 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_6046_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xd: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),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)),Xd)) = aa(nat,word(B),semiring_1_of_nat(word(B)),Xd) ) ) ) ).

% ucast_of_nat_small
tff(fact_6047_bit__last__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(nat,$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_6048_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_6049_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_6050_unat__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(B)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xd)) = modulo_modulo(nat,aa(word(B),nat,semiring_1_unsigned(B,nat),Xd),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_6051_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_6052_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)) = 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_6053_unat__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Xd)) = modulo_modulo(nat,Xd,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_6054_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_6055_ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A)] :
          ( aa(nat,$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)),Xd)),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_6056_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_6057_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_6058_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_6059_upper__trivial,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( ( Xd != aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) )
         => aa(word(A),$o,ord_less(word(A),Xd),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_6060_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,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_6061_bit__word__scast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(B),W),Nb)
              | ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(B,type2(B))),Nb)
                & aa(nat,$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_6062_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_6063_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_6064_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xd)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono_le
tff(fact_6065_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),modulo_modulo(word(A),A2,B2)) = modulo_modulo(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_6066_bit__signed__drop__bit__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),signed_drop_bit(A,M,W)),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),
              $ite(
                ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),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_6067_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,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_6068_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,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)))),bit_se4730199178511100633sh_bit(A,Nb,ring_1_signed(B,A,W))) ) ).

% signed_push_bit_eq
tff(fact_6069_msb0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A),I: nat] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              | aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              & aa(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,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              <=> aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) ) ) ).

% msb0
tff(fact_6070_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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)),Xd),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem
tff(fact_6071_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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)),Xd),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem'
tff(fact_6072_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_6073_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_6074_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less_eq(nat,Y),Xd)
            <=> aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xd)) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_6075_unat__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = 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_6076_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(nat,$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)),Xd),one_one(word(A))))
          <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),one_one(word(A))) = one_one(word(A)) ) ) ) ) ).

% bool_mask'
tff(fact_6077_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)) = 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_6078_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xd))
          & aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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_6079_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(int,$o,ord_less(int,ring_1_signed(A,int,Xd)),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_6080_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ~ ! [N: int] :
              ( ( Xd = aa(int,word(A),ring_1_of_int(word(A)),N) )
             => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),N)
               => ~ aa(int,$o,ord_less(int,N),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_int_cases
tff(fact_6081_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int,Y: int] :
          ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xd)
            & aa(int,$o,ord_less(int,Xd),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
              & aa(int,$o,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
           => ( ( aa(int,word(A),ring_1_of_int(word(A)),Xd) = aa(int,word(A),ring_1_of_int(word(A)),Y) )
            <=> ( Xd = Y ) ) ) ) ) ).

% word_of_int_inj
tff(fact_6082_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xd: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,ord_less(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
             => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xd)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_mono_le'
tff(fact_6083_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 )
        <=> ? [Q5: 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),Q5),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_6084_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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)),Xd),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_mult_lem
tff(fact_6085_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [B2: word(A),F2: word(B)] :
          ( aa(nat,$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)),F2)),B2)
            <=> aa(nat,$o,ord_less(nat,aa(word(B),nat,semiring_1_unsigned(B,nat),F2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_6086_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : ~ aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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_6087_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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_6088_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,F2: word(A)] :
          ( aa(nat,$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,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Nb)
           => aa(word(A),$o,ord_less(word(A),F2),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% unat_ucast_less_no_overflow
tff(fact_6089_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,F2: word(A)] :
          ( aa(nat,$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,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Nb)
          <=> aa(word(A),$o,ord_less(word(A),F2),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% unat_ucast_less_no_overflow_simp
tff(fact_6090_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Nb: nat] :
          ( aa(word(A),$o,ord_less(word(A),P3),aa(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,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & ! [N10: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nb),N10)
               => ( aa(nat,$o,ord_less(nat,N10),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),P3),N10) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_6091_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,P3: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
         => ( ! [N10: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nb),N10)
               => ( aa(nat,$o,ord_less(nat,N10),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),P3),N10) ) )
          <=> aa(word(A),$o,ord_less(word(A),P3),aa(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_6092_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat,M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),Nb)
         => ( aa(word(A),$o,ord_less(word(A),Xd),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,ord_less_eq(nat,M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ) ).

% nth_bounded
tff(fact_6093_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_6094_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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)),Xd),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xd)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_add_lem
tff(fact_6095_uint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = 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_6096_uint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = 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_6097_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,M: int] :
          ( aa(word(A),$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,ord_less_eq(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))))),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_6098_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_6099_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)) = 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_6100_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_6101_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_6102_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,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))))),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_6103_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),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),Nb))
          <=> aa(word(A),$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_6104_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)) = 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_6105_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)) = 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_6106_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),divide_divide(word(A),A2,B2)) = modulo_modulo(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_6107_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_6108_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_6109_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_6110_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)) = 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_6111_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xd)),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)),Xd),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xd)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_mult_lem
tff(fact_6112_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_6113_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)) = 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_6114_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xd: word(A)] :
          ( aa(nat,$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)))
           => ( divide_divide(word(A),modulo_modulo(word(A),Xd,aa(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)) = modulo_modulo(word(A),divide_divide(word(A),Xd,aa(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_6115_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_6116_msb1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xd),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              | aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xd),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              & aa(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,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xd),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              <=> aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xd)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% msb1
tff(fact_6117_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_6118_unat__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = 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_6119_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xd),Y)) = $ite(aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xd)),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),Xd)),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_6120_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_6121_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),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),Xd)),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_6122_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_6123_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_6124_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_le_mono1
tff(fact_6125_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),I)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),J)) ) ) ) ).

% word_add_le_mono2
tff(fact_6126_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_6127_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_6128_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_6129_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_6130_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),ring_1_signed(A,int,Xd)) ) ).

% sint_ge
tff(fact_6131_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_6132_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xd: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xd))
        <=> ! [I2: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I2) = Xd )
                & aa(int,$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_6133_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xd: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xd))
        <=> ~ ? [I2: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I2) = Xd )
                & aa(int,$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_6134_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: int,A2: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R2) = A2 )
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
           => ( aa(int,$o,ord_less(int,R2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),int,semiring_1_unsigned(A,int),A2) = R2 ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_6135_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_6136_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),divide_divide(word(A),K,Xd))
         => 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),Xd))),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_6137_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_6138_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),divide_divide(word(A),K,Xd))
         => 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),Xd))),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_6139_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),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),Xd)),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_6140_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xd))
        <=> 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),Xd))),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_6141_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xd),Y)
         => ? [Z2: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xd),Z2) )
              & aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xd)),aa(word(A),int,semiring_1_unsigned(A,int),Z2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_le_exists'
tff(fact_6142_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: nat,Xd: nat] :
          ( aa(nat,$o,ord_less(nat,P3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xd))
         => ( aa(nat,$o,ord_less(nat,Xd),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),P3)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xd)) ) ) ) ).

% More_Word.of_nat_power
tff(fact_6143_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_6144_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_6145_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,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$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_6146_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,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$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_6147_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat,M: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(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,$o,ord_less_eq(nat,M),Nb)
           => ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,ord_less(word(A),Xd),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_6148_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,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_6149_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))) = 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_6150_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Nb: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(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)))
           => ( ( Xd != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xd),aa(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_6151_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat,Xd: word(A),Y: word(A)] :
          ( aa(nat,$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),$o,ord_less_eq(word(A),Xd),bit_se2239418461657761734s_mask(word(A),M))
           => ( aa(word(A),$o,ord_less_eq(word(A),Y),bit_se2239418461657761734s_mask(word(A),M))
             => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xd),aa(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)) )
               => ( Xd = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_6152_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),divide_divide(word(A),K,Xd))
         => 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),Xd))),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_6153_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),divide_divide(word(A),K,Xd))
         => 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),Xd))),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_6154_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)) )
             => ( bit_se4730199178511100633sh_bit(word(A),Nb,W) != zero_zero(word(A)) ) ) ) ) ) ).

% push_bit_word_eq_nonzero
tff(fact_6155_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Nb)
         => ( signed_drop_bit(A,Nb,W) = $ite(aa(nat,$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_6156_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_lr(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,Nb,Xs))) ).

% n_lists.simps(2)
tff(fact_6157_uint__and__mask__or__full,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),Mask1: word(A),Mask2: int] :
          ( aa(nat,$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 = 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_6158_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W)) ) ).

% sint_greater_eq
tff(fact_6159_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),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)),Xd)) = aa(nat,int,semiring_1_of_nat(int),Xd) ) ) ) ).

% int_eq_sint
tff(fact_6160_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_6161_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_6162_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_6163_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Nb: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),P3),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,ord_less_eq(nat,Nb),N4)
               => ( aa(nat,$o,ord_less(nat,N4),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),P3),N4) ) ) ) ) ) ).

% le_2p_upper_bits
tff(fact_6164_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Nb: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),P3),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,$o,ord_less_eq(nat,Nb),N4)
             => ( aa(nat,$o,ord_less(nat,N4),type_len0_len_of(A,type2(A)))
               => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),P3),N4) ) ) ) ) ).

% le2p_bits_unset
tff(fact_6165_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: nat,Xd: word(A),M: nat,Sz: nat] :
          ( aa(word(A),$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),$o,ord_less(word(A),Xd),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
             => ( aa(word(A),$o,ord_less(word(A),Xd),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) )
                 => aa(word(A),$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)),Xd),aa(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_6166_bit__horner__sum__uint__exp__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ws: list(word(A)),Nb: nat] :
          ( aa(nat,$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,$o,ord_less(nat,divide_divide(nat,Nb,type_len0_len_of(A,type2(A)))),aa(list(word(A)),nat,size_size(list(word(A))),Ws))
            & aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(nat,word(A),nth(word(A),Ws),divide_divide(nat,Nb,type_len0_len_of(A,type2(A))))),modulo_modulo(nat,Nb,type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_6167_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat,Y: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Xd),Y)
         => ( aa(nat,$o,ord_less(nat,Y),type_len0_len_of(A,type2(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))),Xd)) = 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),Xd))),one_one(word(A))) ) ) ) ) ).

% div_power_helper
tff(fact_6168_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))),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,$o,ord_less_eq(nat,M),Nb)
              & aa(nat,$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))),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_6169_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_6170_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,$o,ord_less_eq(nat,A2),B2)
           => aa(int,$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_6171_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: nat] :
          ( aa(nat,$o,ord_less(nat,Xd),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(int,$o,ord_less_eq(int,zero_zero(int)),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),Xd))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_6172_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_6173_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),Xd)
         => ( aa(int,$o,ord_less(int,Xd),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)),Xd)) = Xd ) ) ) ) ).

% sint_of_int_eq
tff(fact_6174_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,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_6175_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_6176_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_le_mono1
tff(fact_6177_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),signed6721504322012087516modulo(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))) ) ).

% smod_word_min
tff(fact_6178_int__word__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int] : ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Xd)) = aa(int,int,minus_minus(int,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),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_6179_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A)] :
          ( aa(word(A),$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),Xd)),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))))
           => ( divide_divide(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xd),Y),Y) = Xd ) ) ) ) ).

% Word.word_div_mult
tff(fact_6180_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_6181_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),A2: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [A5: word(A)] :
                ( aa(word(A),$o,P,A5)
               => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),A5)
                 => ( aa(word(A),$o,ord_less(word(A),A5),aa(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),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A5)) ) ) )
           => ( ! [A5: word(A)] :
                  ( aa(word(A),$o,P,A5)
                 => ( aa(word(A),$o,ord_less(word(A),A5),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$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))),A5))) ) )
             => aa(word(A),$o,P,A2) ) ) ) ) ).

% word_bit_induct
tff(fact_6182_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,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$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_6183_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_6184_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)))))
         => ( 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_6185_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_6186_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 = 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 = 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,bit_se5641148757651400278ts_bit(int,i),Index),
                          $ite(
                            ( aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,i),Shift)),Least)
                            | aa(int,$o,ord_less_eq(int,Overflow),aa(int,int,minus_minus(int,i),Shift)) ),
                            aa(int,word(A),Arbitrary1,i),
                            aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,i),Shift)) ),
                          $ite(
                            ( aa(int,$o,ord_less(int,i),Least)
                            | aa(int,$o,ord_less_eq(int,Overflow),i) ),
                            aa(int,word(A),Arbitrary2,i),
                            aa(int,word(A),ring_1_of_int(word(A)),i) ) ) ) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
tff(fact_6187_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,Us: nat] :
          ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,ord_less_eq(nat,Us),Sz)
           => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Sz),Us)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),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_6188_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Nb: nat,Xd: word(A),A2: word(A)] :
          ( aa(nat,$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)))
         => ( ( Xd = 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),$o,ord_less_eq(word(A),A2),Xd)
             => ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(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),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_6189_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_6190_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,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Nb) ) ) ).

% decr_length_less_iff
tff(fact_6191_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% length_not_greater_eq_2_iff
tff(fact_6192_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_6193_len__num0,axiom,
    ! [Uu: itself(numeral_num0)] : type_len0_len_of(numeral_num0,Uu) = zero_zero(nat) ).

% len_num0
tff(fact_6194_len__num1,axiom,
    ! [Uu: itself(numeral_num1)] : type_len0_len_of(numeral_num1,Uu) = one_one(nat) ).

% len_num1
tff(fact_6195_len__of__finite__1__def,axiom,
    ! [Xd: itself(finite_1)] : type_len0_len_of(finite_1,Xd) = one_one(nat) ).

% len_of_finite_1_def
tff(fact_6196_len__of__finite__3__def,axiom,
    ! [Xd: itself(finite_3)] : type_len0_len_of(finite_3,Xd) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% len_of_finite_3_def
tff(fact_6197_len__of__finite__2__def,axiom,
    ! [Xd: itself(finite_2)] : type_len0_len_of(finite_2,Xd) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% len_of_finite_2_def
tff(fact_6198_len__not__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( type_len0_len_of(A,type2(A)) != zero_zero(nat) ) ) ).

% len_not_eq_0
tff(fact_6199_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_6200_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% two_less_eq_exp_length
tff(fact_6201_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_6202_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),product_Pair(word(A),word(A),divide_divide(word(A),Xd,Y)),modulo_modulo(word(A),Xd,Y)) = $ite(
                aa(word(A),$o,ord_less_eq(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),Xd),Y),aa(word(A),product_prod(word(A),word(A)),product_Pair(word(A),word(A),zero_zero(word(A))),Xd),aa(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),Xd),Y))),
                $let(
                  q: word(A),
                  q:= bit_se4730199178511100633sh_bit(word(A),one_one(nat),signed7115095781618012415divide(word(A),bit_se4197421643247451524op_bit(word(A),one_one(nat),Xd),Y)),
                  $let(
                    r: word(A),
                    r:= aa(word(A),word(A),minus_minus(word(A),Xd),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                    $ite(aa(word(A),$o,ord_less_eq(word(A),Y),r),aa(word(A),product_prod(word(A),word(A)),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)),product_Pair(word(A),word(A),q),r)) ) ) ) ) ) ) ).

% divmod_via_sdivmod
tff(fact_6203_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_lo(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_6204_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_6205_word__sdiv__div0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : signed7115095781618012415divide(word(A),A2,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_sdiv_div0
tff(fact_6206_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_6207_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_6208_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_6209_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_6210_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_6211_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_6212_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_6213_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_6214_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_6215_word__sdiv__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_sdiv_numerals(5)
tff(fact_6216_word__sdiv__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,W))) ) ).

% word_sdiv_numerals_lhs(2)
tff(fact_6217_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_6218_word__sdiv__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Xd),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)),Xd)),ring_1_signed(A,int,one_one(word(A))))) ) ).

% word_sdiv_numerals(7)
tff(fact_6219_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_6220_word__sdiv__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Xd),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)),Xd)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_sdiv_numerals(4)
tff(fact_6221_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_6222_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_6223_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_6224_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F2: fun(int,A),Xd: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F2,Xd))
        <=> ! [I2: int] :
              ( ( ( Xd = aa(int,word(B),ring_1_of_int(word(B)),I2) )
                & aa(int,$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,F2,I2)) ) ) ) ).

% word_int_split
tff(fact_6225_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F2: fun(int,A),Xd: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F2,Xd))
        <=> ~ ? [N6: int] :
                ( ( Xd = aa(int,word(B),ring_1_of_int(word(B)),N6) )
                & aa(int,$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,F2,N6)) ) ) ) ).

% word_int_split_asm
tff(fact_6226_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F2: fun(int,A),I: int] : word_int_case(A,B,F2,aa(int,word(B),ring_1_of_int(word(B)),I)) = aa(int,A,F2,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_6227_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_6228_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_6229_smod__word__mod__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : signed6721504322012087516modulo(word(A),Xd,zero_zero(word(A))) = Xd ) ).

% smod_word_mod_0
tff(fact_6230_smod__word__0__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),Xd) = zero_zero(word(A)) ) ).

% smod_word_0_mod
tff(fact_6231_smod__word__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,zero_zero(word(A))) = W ) ).

% smod_word_zero
tff(fact_6232_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_6233_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_6234_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_6235_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_6236_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_6237_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_6238_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_6239_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_6240_word__smod__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,W))) ) ).

% word_smod_numerals_lhs(2)
tff(fact_6241_word__smod__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_smod_numerals(5)
tff(fact_6242_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_6243_word__smod__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Xd),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)),Xd)),ring_1_signed(A,int,one_one(word(A))))) ) ).

% word_smod_numerals(7)
tff(fact_6244_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_6245_word__smod__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Xd),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)),Xd)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_smod_numerals(4)
tff(fact_6246_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_6247_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_6248_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_6249_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)) = 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_6250_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat,W: word(B)] :
          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)),bit_se4197421643247451524op_bit(word(B),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),Nb),W)),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_6251_slice1__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat] : slice1(B,A,Nb,zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice1_0
tff(fact_6252_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_6253_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_6254_word__m1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),Y),word_pred(A,zero_zero(word(A)))) ) ).

% word_m1_ge
tff(fact_6255_word__not__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,ord_less(word(A),word_pred(A,zero_zero(word(A)))),Y) ) ).

% word_not_simps(2)
tff(fact_6256_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_6257_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_6258_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_6259_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_6260_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_6261_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_6262_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)) = 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_6263_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),$o),aTP_Lamp_ls(nat,fun(nat,fun(word(A),$o)),Nb),M)) = image(nat,word(A),semiring_1_of_nat(word(A)),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_lt(nat,fun(nat,fun(nat,$o)),Nb),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_6264_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),S3) = S3 ) ).

% image_add_0
tff(fact_6265_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: 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_6266_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: 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_6267_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),V: list(B)] : aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),V)) = image(B,A,F2,aa(list(B),set(B),set2(B),V)) ).

% list.set_map
tff(fact_6268_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A2: 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_6269_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : image(A,A,aTP_Lamp_lu(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_6270_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : image(A,A,aTP_Lamp_lu(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_6271_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_6272_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D)
         => ( image(A,A,aa(A,fun(A,A),times_times(A),D),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_6273_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D)
         => ( image(A,A,aTP_Lamp_lv(A,fun(A,A),D),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A2,D),divide_divide(A,B2,D)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_6274_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_6275_nth__image__indices,axiom,
    ! [A: $tType,L: list(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_6276_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),Nb: nat] :
      ( ( A3 = image(nat,A,F2,collect(nat,aTP_Lamp_as(nat,fun(nat,$o),Nb))) )
     => aa(set(A),$o,finite_finite2(A),A3) ) ).

% nat_seg_image_imp_finite
tff(fact_6277_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
    <=> ? [N6: nat,F6: fun(nat,A)] : A3 = image(nat,A,F6,collect(nat,aTP_Lamp_as(nat,fun(nat,$o),N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_6278_image__set,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : image(B,A,F2,aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),Xs)) ).

% image_set
tff(fact_6279_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_6280_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_6281_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L2: list(A),F2: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L2) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,F2,aa(nat,A,nth(A,L),I3)) = aa(A,B,F2,aa(nat,A,nth(A,L2),I3)) ) )
       => ( image(A,B,F2,aa(list(A),set(A),set2(A),L)) = image(A,B,F2,aa(list(A),set(A),set2(A),L2)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6282_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xd: B,L: list(B)] :
      ( member(A,aa(B,A,F2,Xd),image(B,A,F2,aa(list(B),set(B),set2(B),L)))
    <=> ? [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(B),nat,size_size(list(B)),L))
          & ( aa(B,A,F2,aa(nat,B,nth(B,L),I2)) = aa(B,A,F2,Xd) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6283_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Xd: A,Y: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( image(A,A,real_V8093663219630862766scaleR(A,C2),set_or1337092689740270186AtMost(A,Xd,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),Xd),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_6284_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_6285_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_6286_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xd: A,Y: A] :
          image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,Xd,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),Xd),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,ord_less_eq(A,Xd),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),Xd)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_6287_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_6288_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xd: A,Y: A] :
          image(A,A,aTP_Lamp_lw(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,Xd,Y)) = $ite(
            aa(A,$o,ord_less_eq(A,Xd),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),Xd),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),Xd),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_6289_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_lx(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,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_6290_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_ly(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,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_6291_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_lz(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,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_6292_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_ma(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,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,divide_divide(A,A2,M)),C2),aa(A,A,minus_minus(A,divide_divide(A,B2,M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,divide_divide(A,B2,M)),C2),aa(A,A,minus_minus(A,divide_divide(A,A2,M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_6293_word__sp__01,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType] :
      ( ( type_len(D6)
        & 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(D6,one_one(word(D6))) = zero_zero(word(D6)) ) ) ) ).

% word_sp_01
tff(fact_6294_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_6295_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_6296_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)) = 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_6297_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A),B4: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3,B4)
        <=> ( image(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3) = B4 ) ) ) ).

% bij_betw_add
tff(fact_6298_bit__word__roti__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,W: word(A),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(A),W),nat2(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_6299_bij__betw__Suc,axiom,
    ! [M7: set(nat),N2: set(nat)] :
      ( bij_betw(nat,nat,suc,M7,N2)
    <=> ( image(nat,nat,suc,M7) = N2 ) ) ).

% bij_betw_Suc
tff(fact_6300_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: 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_6301_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: 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_6302_word__roti__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : word_roti(A,zero_zero(int),W) = W ) ).

% word_roti_0
tff(fact_6303_word__roti__0_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int] : word_roti(A,Nb,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_roti_0'
tff(fact_6304_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ member(nat,zero_zero(nat),image(nat,nat,suc,A3)) ).

% zero_notin_Suc_image
tff(fact_6305_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set(A)] : ~ member(option(A),none(A),image(A,option(A),some(A),A3)) ).

% None_notin_image_Some
tff(fact_6306_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image(nat,nat,suc,set_ord_lessThan(nat,Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),Nb) ).

% image_Suc_lessThan
tff(fact_6307_image__Suc__atMost,axiom,
    ! [Nb: 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_6308_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_6309_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_6310_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,Nb)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_ord_lessThan(nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_6311_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_atMost(nat,aa(nat,nat,suc,Nb)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_ord_atMost(nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_6312_subset__subseqs,axiom,
    ! [A: $tType,X7: set(A),Xs: list(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),X7),aa(list(A),set(A),set2(A),Xs))
     => member(set(A),X7,image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_6313_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image(int,int,aTP_Lamp_mb(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_6314_word__roti__conv__mod_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,W: word(A)] : word_roti(A,Nb,W) = word_roti(A,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_6315_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image(code_integer,code_integer,aTP_Lamp_mc(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_6316_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = image(nat,int,semiring_1_of_nat(int),set_ord_lessThan(nat,nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_6317_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xd: nat,Y: nat] :
      image(nat,nat,aTP_Lamp_md(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,Xd,Y)) = $ite(
        aa(nat,$o,ord_less(nat,C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,Xd),C2),aa(nat,nat,minus_minus(nat,Y),C2)),
        $ite(aa(nat,$o,ord_less(nat,Xd),Y),insert(nat,zero_zero(nat),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_6318_word__roti__conv__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,W: word(A)] : word_roti(A,Nb,W) = word_roti(A,modulo_modulo(int,Nb,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))),W) ) ).

% word_roti_conv_mod
tff(fact_6319_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),T: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(set(A),set(A),minus_minus(set(A),S),T)) = aa(set(A),set(A),minus_minus(set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),S)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),T)) ) ).

% translation_diff
tff(fact_6320_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(set(A),set(A),uminus_uminus(set(A)),T)) = aa(set(A),set(A),uminus_uminus(set(A)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),T)) ) ).

% translation_Compl
tff(fact_6321_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),Nb: nat] :
      ( vEBT_invar_vebt(S,M)
     => ( ! [X: vEBT_VEBT] :
            ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
           => vEBT_invar_vebt(X,Nb) )
       => ( ( M = aa(nat,nat,suc,Nb) )
         => ( ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,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,X)) = archimedean_ceiling(real,aa(real,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,S)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) )
             => ( aa(nat,int,semiring_1_of_nat(int),lattic643756798349783984er_Max(nat,image(vEBT_VEBT,nat,vEBT_VEBT_height,insert(vEBT_VEBT,S,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))))) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_6322_max__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Nb: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,ord_max(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),lattic643756798349783984er_Max(nat,image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13)))))))) ) ).

% max_idx_list
tff(fact_6323_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( member(vEBT_VEBT,T,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)),lattic643756798349783984er_Max(nat,image(vEBT_VEBT,nat,vEBT_VEBT_height,insert(vEBT_VEBT,Summary,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ) ).

% height_compose_list
tff(fact_6324_max__ins__scaled,axiom,
    ! [Nb: nat,X14: vEBT_VEBT,M: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),lattic643756798349783984er_Max(nat,insert(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14),image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))))) ).

% max_ins_scaled
tff(fact_6325_height__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,ord_max(nat,Foo),lattic643756798349783984er_Max(nat,image(vEBT_VEBT,nat,vEBT_VEBT_height,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))) ) ).

% height_i_max
tff(fact_6326_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ( lattic643756798349783984er_Max(nat,collect(nat,aTP_Lamp_ak(nat,fun(nat,$o),Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_6327_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xd: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less(A,lattic643756798349783984er_Max(A,A3)),Xd)
            <=> ! [X5: A] :
                  ( member(A,X5,A3)
                 => aa(A,$o,ord_less(A,X5),Xd) ) ) ) ) ) ).

% Max_less_iff
tff(fact_6328_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xd: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less(A,Xd),lattic643756798349783984er_Max(A,A3))
            <=> ? [X5: A] :
                  ( member(A,X5,A3)
                  & aa(A,$o,ord_less(A,Xd),X5) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_6329_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( lattic643756798349783984er_Max(A,A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_6330_divide__nat__def,axiom,
    ! [M: nat,Nb: 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_me(nat,fun(nat,fun(nat,$o)),M),Nb)))) ).

% divide_nat_def
tff(fact_6331_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F2: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,image(A,B,aa(B,fun(A,B),aTP_Lamp_mf(fun(A,B),fun(B,fun(A,B)),F2),K),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,image(A,B,F2,S3))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_6332_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,image(vEBT_VEBT,nat,vEBT_VEBT_height,insert(vEBT_VEBT,Summary,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_6333_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xd) = Y )
     => ( ( ? [A5: $o,B3: $o] : Xd = vEBT_Leaf((A5),(B3))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,image(vEBT_VEBT,nat,vEBT_VEBT_height,insert(vEBT_VEBT,Summary2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_6334_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xd) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_height_rel,Xd)
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ( ( Y = zero_zero(nat) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Leaf((A5),(B3))) ) )
         => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,image(vEBT_VEBT,nat,vEBT_VEBT_height,insert(vEBT_VEBT,Summary2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Node(Uu2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_6335_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,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_6336_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero(int) )
     => ( lattic643756798349783984er_Max(int,collect(int,aTP_Lamp_ai(int,fun(int,$o),Nb))) = abs_abs(int,Nb) ) ) ).

% Max_divisors_self_int
tff(fact_6337_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,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_6338_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,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_6339_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,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_6340_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,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_6341_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,one_one(word(A)),zero_zero(word(A)))
      <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(3)
tff(fact_6342_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,zero_zero(word(A)),one_one(word(A)))
      <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(1)
tff(fact_6343_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,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_6344_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,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_6345_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,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_6346_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N: nat] : word_sle(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less_eq(nat,Nb),N3)
           => word_sle(A,aa(nat,word(A),F2,Nb),aa(nat,word(A),F2,N3)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_6347_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N: nat] : word_sle(A,aa(nat,word(A),F2,aa(nat,nat,suc,N)),aa(nat,word(A),F2,N))
         => ( aa(nat,$o,ord_less_eq(nat,Nb),N3)
           => word_sle(A,aa(nat,word(A),F2,N3),aa(nat,word(A),F2,Nb)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_6348_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xd),aa(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)),Xd) ) ) ).

% word_0_sle_from_less
tff(fact_6349_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_6350_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_6351_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_6352_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_6353_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_6354_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_6355_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_6356_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_6357_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_6358_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_6359_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N: nat] : word_sless(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less(nat,Nb),N3)
           => word_sless(A,aa(nat,word(A),F2,Nb),aa(nat,word(A),F2,N3)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_6360_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Nb: nat,M: nat] :
          ( ! [N: nat] : word_sless(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( word_sless(A,aa(nat,word(A),F2,Nb),aa(nat,word(A),F2,M))
          <=> aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_6361_word__sless__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( word_sless(A,A2,B2)
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2)) ) ) ).

% word_sless_alt
tff(fact_6362_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( word_sless(A,Xd,Y)
         => aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,Xd)),aa(int,int,minus_minus(int,ring_1_signed(A,int,Y)),one_one(int))) ) ) ).

% word_sless_sint_le
tff(fact_6363_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_6364_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_6365_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_6366_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_6367_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),ord_min(A,Xd,Y))
        <=> ( aa(A,$o,ord_less(A,Z),Xd)
            & aa(A,$o,ord_less(A,Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_6368_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_6369_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_6370_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_6371_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_6372_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_6373_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_6374_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_6375_min__0R,axiom,
    ! [Nb: nat] : ord_min(nat,Nb,zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_6376_min__0L,axiom,
    ! [Nb: nat] : ord_min(nat,zero_zero(nat),Nb) = zero_zero(nat) ).

% min_0L
tff(fact_6377_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_6378_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_6379_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,ord_min(nat,M,Nb)),A2) ) ).

% take_bit_take_bit
tff(fact_6380_signed__take__bit__signed__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_ri4674362597316999326ke_bit(A,Nb),A2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,ord_min(nat,M,Nb)),A2) ) ).

% signed_take_bit_signed_take_bit
tff(fact_6381_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : ord_min(A,zero_zero(A),aa(num,A,numeral_numeral(A),Xd)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_6382_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : ord_min(A,aa(num,A,numeral_numeral(A),Xd),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_6383_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(1)
tff(fact_6384_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_6385_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_6386_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : ord_min(A,one_one(A),aa(num,A,numeral_numeral(A),Xd)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_6387_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: num] : ord_min(A,aa(num,A,numeral_numeral(A),Xd),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_6388_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_6389_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_6390_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_6391_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_6392_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_6393_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_6394_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se2239418461657761734s_mask(A,Nb)) = bit_se2239418461657761734s_mask(A,ord_min(nat,M,Nb)) ) ).

% take_bit_of_mask
tff(fact_6395_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(2)
tff(fact_6396_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(3)
tff(fact_6397_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(4)
tff(fact_6398_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_6399_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_6400_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_6401_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_6402_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xd: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,ord_max(A,Xd),Y)) = ord_min(A,aa(A,A,uminus_uminus(A),Xd),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_6403_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xd: A,Y: A] : aa(A,A,uminus_uminus(A),ord_min(A,Xd,Y)) = aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),Xd)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_6404_concat__bit__assoc__sym,axiom,
    ! [M: nat,Nb: nat,K: int,L: int,R2: int] : aa(int,int,bit_concat_bit(M,aa(int,int,bit_concat_bit(Nb,K),L)),R2) = aa(int,int,bit_concat_bit(ord_min(nat,M,Nb),K),aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,M),Nb),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_6405_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_6406_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_6407_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),ord_min(A,Y,Z)) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Z)) ) ).

% min_add_distrib_right
tff(fact_6408_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),ord_min(A,Xd,Y)),Z) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Z),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_6409_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,Xd,Y)),Z)
        <=> ( aa(A,$o,ord_less(A,Xd),Z)
            | aa(A,$o,ord_less(A,Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_6410_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_6411_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_6412_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_6413_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_6414_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xd: A,Y: A,Z: A] : aa(A,A,minus_minus(A,ord_min(A,Xd,Y)),Z) = ord_min(A,aa(A,A,minus_minus(A,Xd),Z),aa(A,A,minus_minus(A,Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_6415_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_6416_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa: A] :
          ord_min(A,X4,Xa) = $ite(aa(A,$o,ord_less_eq(A,X4),Xa),X4,Xa) ) ).

% min_def_raw
tff(fact_6417_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xd: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),ord_min(nat,Xd,Y)) = ord_min(A,aa(nat,A,semiring_1_of_nat(A),Xd),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_6418_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,M),aa(int,int,bit_concat_bit(Nb,K),L)) = aa(int,int,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_6419_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),ord_min(A,Xd,Y)),P3) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P3),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),P3),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% min_mult_distrib_right
tff(fact_6420_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xd: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,ord_max(A,Xd),Y)),P3) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P3),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xd),P3),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% max_mult_distrib_right
tff(fact_6421_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xd: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),ord_min(A,Xd,Y)) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P3),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xd),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% min_mult_distrib_left
tff(fact_6422_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xd: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,ord_max(A,Xd),Y)) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P3),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xd)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xd),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% max_mult_distrib_left
tff(fact_6423_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A,P3: A] :
          divide_divide(A,ord_min(A,Xd,Y),P3) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P3),ord_min(A,divide_divide(A,Xd,P3),divide_divide(A,Y,P3)),aa(A,A,ord_max(A,divide_divide(A,Xd,P3)),divide_divide(A,Y,P3))) ) ).

% min_divide_distrib_right
tff(fact_6424_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xd: A,Y: A,P3: A] :
          divide_divide(A,aa(A,A,ord_max(A,Xd),Y),P3) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P3),aa(A,A,ord_max(A,divide_divide(A,Xd,P3)),divide_divide(A,Y,P3)),ord_min(A,divide_divide(A,Xd,P3),divide_divide(A,Y,P3))) ) ).

% max_divide_distrib_right
tff(fact_6425_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Nb: nat] : modulo_modulo(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)) = 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_6426_mod__mod__power,axiom,
    ! [K: nat,M: nat,Nb: nat] : modulo_modulo(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)) = 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_6427_Word_Obit__mask__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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_6428_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat,M: nat] : 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_6429_bit__slice__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(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,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_6430_bit__slice1__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),slice1(B,A,M,W)),Nb)
        <=> ( aa(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,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_6431_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_6432_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),Bs)),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($o)),Bs)))
            & aa(nat,$o,nth($o,Bs),Nb) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_6433_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] :
          bit_se4197421643247451524op_bit(A,M,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,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_6434_UNIV__word__eq__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(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_6435_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_min(extended_enat,Q2,zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_6436_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_min(extended_enat,zero_zero(extended_enat),Q2) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_6437_possible__bit__min,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,ord_min(nat,I,J))
        <=> ( bit_se6407376104438227557le_bit(A,Tyrep,I)
            | bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_min
tff(fact_6438_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: 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_6439_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: 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_6440_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_6441_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_6442_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_6443_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,I)
         => ( aa(nat,$o,ord_less_eq(nat,J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_6444_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat)) ) ).

% possible_bit_0
tff(fact_6445_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = insert(option(A),none(A),image(A,option(A),some(A),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_6446_bit__eqI,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ! [N: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) ) )
         => ( A2 = B2 ) ) ) ).

% bit_eqI
tff(fact_6447_bit__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ! [N6: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N6)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N6)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N6) ) ) ) ) ).

% bit_eq_iff
tff(fact_6448_impossible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,A2: A] :
          ( ~ bit_se6407376104438227557le_bit(A,type2(A),Nb)
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% impossible_bit
tff(fact_6449_bit__imp__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
         => bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% bit_imp_possible_bit
tff(fact_6450_notin__range__Some,axiom,
    ! [A: $tType,Xd: option(A)] :
      ( ~ member(option(A),Xd,image(A,option(A),some(A),top_top(set(A))))
    <=> ( Xd = none(A) ) ) ).

% notin_range_Some
tff(fact_6451_bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_not_iff
tff(fact_6452_semiring__bit__operations__class_Obit__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            | ( ( M = Nb )
              & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ) ).

% semiring_bit_operations_class.bit_set_bit_iff
tff(fact_6453_bit__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se8732182000553998342ip_bit(A,M,A2)),Nb)
        <=> ( ( ( M = Nb )
            <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) )
            & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ).

% bit_flip_bit_iff
tff(fact_6454_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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_6455_bit__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(int,A,ring_1_of_int(A),K)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ) ).

% bit_of_int_iff
tff(fact_6456_bit__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A2)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),ord_min(nat,M,Nb)) ) ) ) ).

% bit_signed_take_bit_iff
tff(fact_6457_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,M),Nb) ) ) ) ).

% bit_of_nat_iff
tff(fact_6458_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_6459_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A2)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,A2),one_one(A))),Nb) ) ) ) ).

% bit_minus_iff
tff(fact_6460_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xd: 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)),Xd),Y)),
              $ite(aa(word(A),$o,ord_less(word(A),Xd),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = ord_min(word(A),Xd,Y) ) ).

% bit_twiddle_min
tff(fact_6461_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se4730199178511100633sh_bit(A,M,A2)),Nb)
        <=> ( aa(nat,$o,ord_less_eq(nat,M),Nb)
            & bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ) ).

% bit_push_bit_iff
tff(fact_6462_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_6463_bit__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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_6464_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$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_6465_bit__not__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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_6466_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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,ord_less_eq(nat,M),Nb) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6467_range__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( 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_6468_UNIV__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(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_6469_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)))
       => ( image(word(A),word(B),semiring_1_unsigned(A,word(B)),top_top(set(word(A)))) = collect(word(B),aTP_Lamp_mg(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_6470_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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_6471_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$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,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_6472_bit__signed__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,ring_1_signed(B,A,W)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(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_6473_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_6474_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)) = aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),modulo_modulo(nat,Nb,type_len0_len_of(A,type2(A)))),bit_se4197421643247451524op_bit(int,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),modulo_modulo(nat,Nb,type_len0_len_of(A,type2(A)))),W))) ) ).

% uint_word_rotr_eq
tff(fact_6475_range__mult,axiom,
    ! [A2: real] :
      image(real,real,aa(real,fun(real,real),times_times(real),A2),top_top(set(real))) = $ite(A2 = zero_zero(real),insert(real,zero_zero(real),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_6476_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_mh(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_6477_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_mh(nat,$o)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_6478_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ aa(set(list(A)),$o,finite_finite2(list(A)),top_top(set(list(A)))) ).

% infinite_UNIV_listI
tff(fact_6479_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_6480_bit__set__bits__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Nb: nat] :
          ( aa(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_6481_word__type__copy__misc_Oset__bits__code,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),P: fun(nat,$o)] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( bit_bi4170147762399595738t_bits(B,P) = aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,P),Size),zero_zero(B)) ) ) ) ).

% word_type_copy_misc.set_bits_code
tff(fact_6482_range__mod,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( image(nat,nat,aTP_Lamp_mi(nat,fun(nat,nat),Nb),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_6483_set__bits__conv__set__bits__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),F2) = code_T2661198915054445665ts_aux(A,F2,type_len0_len_of(A,type2(A)),zero_zero(word(A))) ) ).

% set_bits_conv_set_bits_aux
tff(fact_6484_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),bit_bi4170147762399595738t_bits(word(A),F2)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,F2,Nb) ) ) ) ).

% word_test_bit_set_bits
tff(fact_6485_bit__word__rotr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(A),W),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_6486_UNIV__nat__eq,axiom,
    top_top(set(nat)) = insert(nat,zero_zero(nat),image(nat,nat,suc,top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_6487_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,W: word(A)] :
          word_roti(A,I,W) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),I),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_6488_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_mh(nat,$o)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_6489_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_mj(nat,$o)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_set_bits_K_True
tff(fact_6490_word__rotx__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [I: nat] :
          ( ( aa(word(A),word(A),word_rotr(A,I),zero_zero(word(A))) = zero_zero(word(A)) )
          & ( aa(word(B),word(B),word_rotl(B,I),zero_zero(word(B))) = zero_zero(word(B)) ) ) ) ).

% word_rotx_0
tff(fact_6491_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))),modulo_modulo(nat,Nb,type_len0_len_of(A,type2(A))))) ) ).

% word_rotl_eq_word_rotr
tff(fact_6492_bit__word__rotl__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Nb: nat] :
          ( aa(nat,$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,bit_se5641148757651400278ts_bit(word(A),W),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))),modulo_modulo(nat,M,type_len0_len_of(A,type2(A))))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotl_iff
tff(fact_6493_root__def,axiom,
    ! [Nb: nat,Xd: real] :
      aa(real,real,root(Nb),Xd) = $ite(Nb = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_mk(nat,fun(real,real),Nb),Xd)) ).

% root_def
tff(fact_6494_word__roti_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] : word_roti(A,Xaa,word2(A,Xd)) = word2(A,aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),nat2(modulo_modulo(int,Xaa,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),bit_se4197421643247451524op_bit(int,nat2(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))),Xd))),aa(int,int,bit_se2584673776208193580ke_bit(int,nat2(modulo_modulo(int,Xaa,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),Xd))) ) ).

% word_roti.abs_eq
tff(fact_6495_one__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = word2(A,one_one(int)) ) ) ).

% one_word_def
tff(fact_6496_zero__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = word2(A,zero_zero(int)) ) ) ).

% zero_word_def
tff(fact_6497_set__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: nat,Xd: int] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se5668285175392031749et_bit(word(A)),Xaa),word2(A,Xd)) = word2(A,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xaa),Xd)) ) ).

% set_bit_word.abs_eq
tff(fact_6498_word__succ_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int] : word_succ(A,word2(A,Xd)) = word2(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),one_one(int))) ) ).

% word_succ.abs_eq
tff(fact_6499_word__pred_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int] : word_pred(A,word2(A,Xd)) = word2(A,aa(int,int,minus_minus(int,Xd),one_one(int))) ) ).

% word_pred.abs_eq
tff(fact_6500_less__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] :
          ( aa(word(A),$o,ord_less(word(A),word2(A,Xaa)),word2(A,Xd))
        <=> aa(int,$o,ord_less(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))),Xd)) ) ) ).

% less_word.abs_eq
tff(fact_6501_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int,X4: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),word2(A,Xd)),X4)
        <=> ( aa(nat,$o,ord_less(nat,X4),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,Xd),X4) ) ) ) ).

% bit_word.abs_eq
tff(fact_6502_divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] : divide_divide(word(A),word2(A,Xaa),word2(A,Xd)) = word2(A,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))),Xd))) ) ).

% divide_word.abs_eq
tff(fact_6503_modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] : modulo_modulo(word(A),word2(A,Xaa),word2(A,Xd)) = word2(A,modulo_modulo(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))),Xd))) ) ).

% modulo_word.abs_eq
tff(fact_6504_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] :
          ( word_sle(A,word2(A,Xaa),word2(A,Xd))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),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)))),Xd)) ) ) ).

% word_sle.abs_eq
tff(fact_6505_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] :
          ( word_sless(A,word2(A,Xaa),word2(A,Xd))
        <=> aa(int,$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)))),Xd)) ) ) ).

% word_sless.abs_eq
tff(fact_6506_signed_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Xd: int] : ring_1_signed(B,A,word2(B,Xd)) = 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)))),Xd)) ) ).

% signed.abs_eq
tff(fact_6507_signed__drop__bit_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: nat,Xd: int] : signed_drop_bit(A,Xaa,word2(A,Xd)) = word2(A,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)))),Xd))) ) ).

% signed_drop_bit.abs_eq
tff(fact_6508_signed__modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] : signed6721504322012087516modulo(word(A),word2(A,Xaa),word2(A,Xd)) = 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)))),Xd))) ) ).

% signed_modulo_word.abs_eq
tff(fact_6509_signed__divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xd: int] : signed7115095781618012415divide(word(A),word2(A,Xaa),word2(A,Xd)) = 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)))),Xd))) ) ).

% signed_divide_word.abs_eq
tff(fact_6510_word__rotr_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: nat,Xd: int] : aa(word(A),word(A),word_rotr(A,Xaa),word2(A,Xd)) = word2(A,aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),modulo_modulo(nat,Xaa,type_len0_len_of(A,type2(A)))),bit_se4197421643247451524op_bit(int,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))),Xd))),aa(int,int,bit_se2584673776208193580ke_bit(int,modulo_modulo(nat,Xaa,type_len0_len_of(A,type2(A)))),Xd))) ) ).

% word_rotr.abs_eq
tff(fact_6511_word__rotl_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: nat,Xd: int] : aa(word(A),word(A),word_rotl(A,Xaa),word2(A,Xd)) = word2(A,aa(int,int,bit_concat_bit(modulo_modulo(nat,Xaa,type_len0_len_of(A,type2(A))),bit_se4197421643247451524op_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),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))),Xd))),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),modulo_modulo(nat,Xaa,type_len0_len_of(A,type2(A))))),Xd))) ) ).

% word_rotl.abs_eq
tff(fact_6512_signed__cast_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xd: int] : signed_cast(B,A,word2(B,Xd)) = 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)))),Xd)) ) ).

% signed_cast.abs_eq
tff(fact_6513_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int] : the_signed_int(A,word2(A,Xd)) = 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)))),Xd) ) ).

% the_signed_int.abs_eq
tff(fact_6514_bin__last__set__bits,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,F2))
      <=> aa(nat,$o,F2,zero_zero(nat)) ) ) ).

% bin_last_set_bits
tff(fact_6515_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_6516_wf__set__bits__int__Suc,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_ml(fun(nat,$o),fun(nat,$o),F2))
    <=> bit_wf_set_bits_int(F2) ) ).

% wf_set_bits_int_Suc
tff(fact_6517_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_6518_of__nat__CHAR,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).

% of_nat_CHAR
tff(fact_6519_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X) = zero_zero(A) )
               => aa(nat,$o,dvd_dvd(nat,C2),X) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_6520_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Nb) = zero_zero(A) )
        <=> aa(nat,$o,dvd_dvd(nat,semiri4206861660011772517g_char(A,type2(A))),Nb) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_6521_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_6522_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) )
           => ( ! [X: nat] :
                  ( aa(nat,$o,ord_less(nat,zero_zero(nat)),X)
                 => ( aa(nat,$o,ord_less(nat,X),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_6523_int__set__bits__unfold__BIT,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( bit_bi4170147762399595738t_bits(int,F2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,F2,zero_zero(nat)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F2,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_6524_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),bit_Sh8784991116023147202shiftr(A,W,M)),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),
              $ite(
                ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),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_6525_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : comp(list(C),list(B),list(A),map(C,B,F2),map(A,C,G)) = map(A,B,comp(C,B,A,F2,G)) ).

% map_comp_map
tff(fact_6526_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F2),aa(list(C),list(B),map(C,B,G),Xs)) = aa(list(C),list(A),map(C,A,comp(B,A,C,F2,G)),Xs) ).

% map_map
tff(fact_6527_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),List: list(C)] : aa(list(B),list(A),map(B,A,F2),aa(list(C),list(B),map(C,B,G),List)) = aa(list(C),list(A),map(C,A,comp(B,A,C,F2,G)),List) ).

% List.map.compositionality
tff(fact_6528_list_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,A),F2: fun(C,B),V: list(C)] : aa(list(B),list(A),map(B,A,G),aa(list(C),list(B),map(C,B,F2),V)) = aa(list(C),list(A),map(C,A,comp(B,A,C,G,F2)),V) ).

% list.map_comp
tff(fact_6529_List_Omap_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : comp(list(C),list(B),list(A),map(C,B,F2),map(A,C,G)) = map(A,B,comp(C,B,A,F2,G)) ).

% List.map.comp
tff(fact_6530_sshiftr__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : bit_Sh8784991116023147202shiftr(A,W,zero_zero(nat)) = W ) ).

% sshiftr_of_0
tff(fact_6531_sshiftr__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : bit_Sh8784991116023147202shiftr(A,zero_zero(word(A)),Nb) = zero_zero(word(A)) ) ).

% sshiftr_0
tff(fact_6532_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_6533_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_6534_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_6535_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_6536_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_6537_bin__rest__set__bits,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( divide_divide(int,bit_bi4170147762399595738t_bits(int,F2),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F2,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_6538_foldr_Ofoldr__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xd: B,Xs: list(B)] : foldr(B,A,F2,aa(list(B),list(B),cons(B,Xd),Xs)) = comp(A,A,A,aa(B,fun(A,A),F2,Xd),foldr(B,A,F2,Xs)) ).

% foldr.foldr_Cons
tff(fact_6539_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) )
         => ( ! [X: B,Y2: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X)),aa(B,A,H,Y2))
           => ( 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_6540_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_6541_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_6542_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5641148757651400278ts_bit(A,bit_se4197421643247451524op_bit(A,Nb,A2)) = comp(nat,$o,nat,bit_se5641148757651400278ts_bit(A,A2),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ).

% bit_drop_bit_eq
tff(fact_6543_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_6544_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_6545_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_6546_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_6547_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_6548_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_6549_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,comp(A,A,nat,inverse_inverse(A),F2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_mm(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_6550_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X: A,Y2: A] :
                ( member(A,X,A3)
               => ( member(A,Y2,A3)
                 => ( ( X != Y2 )
                   => ( ( aa(A,B,H,X) = aa(A,B,H,Y2) )
                     => ( aa(B,C,G,aa(A,B,H,X)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),image(A,B,H,A3)) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,comp(B,C,A,G,H)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_6551_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X: A,Y2: A] :
                ( member(A,X,A3)
               => ( member(A,Y2,A3)
                 => ( ( X != Y2 )
                   => ( ( aa(A,B,H,X) = aa(A,B,H,Y2) )
                     => ( aa(B,C,G,aa(A,B,H,X)) = one_one(C) ) ) ) ) )
           => ( groups7121269368397514597t_prod(B,C,G,image(A,B,H,A3)) = groups7121269368397514597t_prod(A,C,comp(B,C,A,G,H),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_6552_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(C,B,G,aa(A,C,F2,I3))) )
           => aa(B,$o,ord_less_eq(B,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),image(A,C,F2,I6))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,comp(C,B,A,G,F2)),I6)) ) ) ) ).

% sum_image_le
tff(fact_6553_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_6554_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_6555_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_6556_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_6557_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_6558_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_6559_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_lj(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_6560_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_lj(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_6561_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_lj(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_6562_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_lj(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_6563_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_6564_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_6565_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_6566_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_6567_shiftl__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_Sh4282982442137083160shiftl(A,zero_zero(A),Nb) = zero_zero(A) ) ).

% shiftl_0
tff(fact_6568_shiftl__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083160shiftl(A,A2,zero_zero(nat)) = A2 ) ).

% shiftl_of_0
tff(fact_6569_size__list__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F2),aa(list(B),list(A),map(B,A,G),Xs)) = aa(list(B),nat,size_list(B,comp(A,nat,B,F2,G)),Xs) ).

% size_list_map
tff(fact_6570_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)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_Suc
tff(fact_6571_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)) = bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),Nb),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_numeral
tff(fact_6572_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)) = 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_6573_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)) = 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_6574_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_6575_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_6576_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F2: fun(C,B),Xs: list(C),A2: A] : aa(A,A,foldr(B,A,G,aa(list(C),list(B),map(C,B,F2),Xs)),A2) = aa(A,A,foldr(C,A,comp(B,fun(A,A),C,G,F2),Xs),A2) ).

% foldr_map
tff(fact_6577_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : comp(list(B),nat,list(A),size_list(B,F2),map(A,B,G)) = size_list(A,comp(B,nat,A,F2,G)) ).

% list.size_gen_o_map
tff(fact_6578_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_6579_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_6580_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_6581_one__bit__shiftl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : generi7602027413899671122et_bit(word(A),zero_zero(word(A)),Nb,$true) = bit_Sh4282982442137083160shiftl(word(A),one_one(word(A)),Nb) ) ).

% one_bit_shiftl
tff(fact_6582_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xd: A,Nb: nat] : bit_Sh4282982442137083160shiftl(A,Xd,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xd),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_6583_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_6584_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)) = 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_6585_shiftr__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_Sh4282982442137083166shiftr(A,zero_zero(A),Nb) = zero_zero(A) ) ).

% shiftr_0
tff(fact_6586_shiftr__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083166shiftr(A,A2,zero_zero(nat)) = A2 ) ).

% shiftr_of_0
tff(fact_6587_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_6588_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)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_Suc
tff(fact_6589_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)) = bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),Nb),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_numeral
tff(fact_6590_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_6591_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)) = 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_6592_bit__shiftr__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : bit_se5641148757651400278ts_bit(A,bit_Sh4282982442137083166shiftr(A,A2,Nb)) = comp(nat,$o,nat,bit_se5641148757651400278ts_bit(A,A2),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ).

% bit_shiftr_eq
tff(fact_6593_shiftr__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xd: A,Nb: nat] : bit_Sh4282982442137083166shiftr(A,Xd,Nb) = divide_divide(A,Xd,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_6594_bit__revcast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),revcast(B,A,W)),Nb)
        <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),Nb)
            & aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(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_6595_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,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_6596_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_6597_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,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_6598_word__sless__msb__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A),Y: word(A)] :
          ( word_sless(A,Xd,Y)
        <=> ( ( most_s684356279273892711sb_msb(word(A),Y)
             => most_s684356279273892711sb_msb(word(A),Xd) )
            & ( ( most_s684356279273892711sb_msb(word(A),Xd)
                & ~ most_s684356279273892711sb_msb(word(A),Y) )
              | aa(word(A),$o,ord_less(word(A),Xd),Y) ) ) ) ) ).

% word_sless_msb_less
tff(fact_6599_msb__word__iff__sless__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> word_sless(A,W,zero_zero(word(A))) ) ) ).

% msb_word_iff_sless_0
tff(fact_6600_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_6601_word__msb__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ~ most_s684356279273892711sb_msb(word(A),zero_zero(word(A))) ) ).

% word_msb_0
tff(fact_6602_word__msb__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),zero_zero(int)) ) ) ).

% word_msb_sint
tff(fact_6603_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_6604_msb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$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_6605_msb__word__iff__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$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_6606_word__msb__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$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_6607_msb__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: int] :
          ( most_s684356279273892711sb_msb(word(A),aa(int,word(A),ring_1_of_int(word(A)),Xd))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,Xd),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% msb_word_of_int
tff(fact_6608_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_6609_word__sint__msb__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] :
          ring_1_signed(A,int,Xd) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xd)),
            $ite(most_s684356279273892711sb_msb(word(A),Xd),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)),Xd)),zero_zero(int))) ) ).

% word_sint_msb_eq
tff(fact_6610_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),A2)
        <=> aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),A2) ) ) ).

% msb_big
tff(fact_6611_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_6612_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_6613_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,F2),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_6614_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_6615_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_mn(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6616_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_6617_inj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A))))
    <=> inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_map
tff(fact_6618_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_6619_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_mo(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_6620_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,ord_less(A,X),Y2)
             => ( member(A,X,A3)
               => ( member(A,Y2,A3)
                 => ( aa(A,B,F2,X) != aa(A,B,F2,Y2) ) ) ) )
         => ( ! [X: A,Y2: A] :
                ( member(A,X,A3)
               => ( member(A,Y2,A3)
                 => ( aa(A,$o,ord_less_eq(A,X),Y2)
                    | aa(A,$o,ord_less_eq(A,Y2),X) ) ) )
           => inj_on(A,B,F2,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_6621_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_6622_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B4)
     => ( aa(set(A),$o,ord_less(set(A),A3),B4)
       => aa(set(B),$o,ord_less(set(B),image(A,B,F2,A3)),image(A,B,F2,B4)) ) ) ).

% inj_on_strict_subset
tff(fact_6623_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_6624_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_mp(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_6625_map__injective,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),map(B,A,F2),Ys) )
     => ( inj_on(B,A,F2,top_top(set(B)))
       => ( Xs = Ys ) ) ) ).

% map_injective
tff(fact_6626_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A))))
     => inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_mapD
tff(fact_6627_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_6628_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,ord_less(A,X),Y2)
             => ( aa(A,B,F2,X) != aa(A,B,F2,Y2) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_6629_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_6630_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [N: nat,F4: fun(nat,A)] :
          ( ( A3 = image(nat,A,F4,collect(nat,aTP_Lamp_as(nat,fun(nat,$o),N))) )
          & inj_on(nat,A,F4,collect(nat,aTP_Lamp_as(nat,fun(nat,$o),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6631_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xd: real,D3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( Xd != zero_zero(real) )
       => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
             => ( D3 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
             => ( aa(real,$o,ord_less(real,Xd),zero_zero(real))
               => ( D3 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
               => ( D3 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Nb),D3,topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_6632_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xd: 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,Xd),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),Xd)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_6633_inj__on__Cons1,axiom,
    ! [A: $tType,Xd: A,A3: set(list(A))] : inj_on(list(A),list(A),cons(A,Xd),A3) ).

% inj_on_Cons1
tff(fact_6634_inj__on__diff__nat,axiom,
    ! [N2: set(nat),K: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N2)
         => aa(nat,$o,ord_less_eq(nat,K),N) )
     => inj_on(nat,nat,aTP_Lamp_md(nat,fun(nat,nat),K),N2) ) ).

% inj_on_diff_nat
tff(fact_6635_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,collect(set(nat),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_6636_inj__Suc,axiom,
    ! [N2: set(nat)] : inj_on(nat,nat,suc,N2) ).

% inj_Suc
tff(fact_6637_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xd: A,S: set(A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xd),image(A,A,G,S)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xd,S))
           => has_field_derivative(A,comp(A,A,A,F2,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_image_chain
tff(fact_6638_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xd: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xd),image(A,A,aa(A,fun(A,A),plus_plus(A),Z),S3)))
         => has_field_derivative(A,comp(A,A,A,F2,aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xd,S3)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_6639_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xd: A,Db: A,S: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xd),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xd,S))
           => has_field_derivative(A,comp(A,A,A,F2,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_chain
tff(fact_6640_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xd: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xd,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_mq(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),aa(A,A,G,Xd))),M),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_6641_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X,top_top(set(real))))
       => ( aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,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_6642_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S: set(A),G: fun(A,A),G2: fun(A,A),F2: fun(A,A),F7: A,Xd: A] :
          ( ! [X: A] :
              ( member(A,X,S)
             => has_field_derivative(A,G,aa(A,A,G2,X),topolo174197925503356063within(A,X,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,Xd,top_top(set(A))))
           => ( member(A,aa(A,A,F2,Xd),S)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_mr(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F7),aa(A,A,G2,aa(A,A,F2,Xd))),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_6643_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G2: fun(A,A),F2: fun(A,A),F7: A,Xd: A] :
          ( ! [X: A] : has_field_derivative(A,G,aa(A,A,G2,X),topolo174197925503356063within(A,X,top_top(set(A))))
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,Xd,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_mr(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F7),aa(A,A,G2,aa(A,A,F2,Xd))),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_6644_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xd: A,Db: A,S: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xd),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xd,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_mr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_chain2
tff(fact_6645_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,aa(A,A,F2,Xd),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ms(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E4),D3),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_chain'
tff(fact_6646_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F7: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,ord_less_eq(real,A2),X)
           => ( aa(real,$o,ord_less_eq(real,X),B2)
             => has_field_derivative(real,F2,aa(real,real,F7,X),topolo174197925503356063within(real,X,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( aa(real,$o,ord_less(real,A2),Z2)
            & aa(real,$o,ord_less(real,Z2),B2)
            & ( aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),aa(real,real,F7,Z2)) ) ) ) ) ).

% MVT2
tff(fact_6647_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xd: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xd,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_mt(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,Xd))),M),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_6648_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xd: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D2)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,Xd)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),H6))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_6649_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xd: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D2)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),H6))),aa(real,real,F2,Xd)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_6650_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Xd: A,Z: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_mu(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_6651_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X,top_top(set(real))))
       => ( divide_divide(real,aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)),aa(real,real,minus_minus(real,B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_6652_DERIV__ln,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xd),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_6653_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun(real,real),G2: fun(real,real)] :
      ( ! [X: real] :
          ( member(real,X,set_or1337092689740270186AtMost(real,A2,B2))
         => has_field_derivative(real,G,aa(real,real,G2,X),topolo174197925503356063within(real,X,top_top(set(real)))) )
     => ( ! [X: real] :
            ( member(real,X,set_or1337092689740270186AtMost(real,A2,B2))
           => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,G2,X)) )
       => ( aa(real,$o,ord_less_eq(real,A2),B2)
         => aa(real,$o,ord_less_eq(real,aa(real,real,G,A2)),aa(real,real,G,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_6654_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,ord_less_eq(real,A2),X)
           => ( aa(real,$o,ord_less_eq(real,X),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,ord_less(real,Y3),zero_zero(real)) ) ) )
       => aa(real,$o,ord_less(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_6655_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,ord_less_eq(real,A2),X)
           => ( aa(real,$o,ord_less_eq(real,X),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,ord_less(real,zero_zero(real)),Y3) ) ) )
       => aa(real,$o,ord_less(real,aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_6656_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less_eq(real,A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,ord_less_eq(real,A2),X)
           => ( aa(real,$o,ord_less_eq(real,X),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,ord_less_eq(real,Y3),zero_zero(real)) ) ) )
       => aa(real,$o,ord_less_eq(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_6657_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less_eq(real,A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,ord_less_eq(real,A2),X)
           => ( aa(real,$o,ord_less_eq(real,X),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,ord_less_eq(real,zero_zero(real)),Y3) ) ) )
       => aa(real,$o,ord_less_eq(real,aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_6658_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,Xd: real,D: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
       => ( ! [Y2: real] :
              ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xd),Y2))),D)
             => ( aa(real,real,F2,Xd) = aa(real,real,F2,Y2) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_6659_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xd: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D2)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,minus_minus(real,Xd),H6))),aa(real,real,F2,Xd)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_6660_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xd: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D2)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,Xd)),aa(real,real,F2,aa(real,real,minus_minus(real,Xd),H6))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_6661_DERIV__isconst__all,axiom,
    ! [F2: fun(real,real),Xd: real,Y: real] :
      ( ! [X: real] : has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,real,F2,Xd) = aa(real,real,F2,Y) ) ) ).

% DERIV_isconst_all
tff(fact_6662_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_mv(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_6663_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xd: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xd,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_mw(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,Xd)))),M),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_6664_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xd: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xd),image(A,A,aa(A,fun(A,A),plus_plus(A),Z),S3)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_mx(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xd,S3)) ) ) ).

% DERIV_at_within_shift
tff(fact_6665_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => ( ( aa(A,A,F2,Xd) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_my(fun(A,A),fun(A,A),F2),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,F2,Xd))),D3)),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xd)))),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_6666_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xd,S))
           => ( ( aa(A,A,G,Xd) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_mz(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G,Xd))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xd)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xd)),aa(A,A,G,Xd))),topolo174197925503356063within(A,Xd,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_6667_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_na(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D3),C2),topolo174197925503356063within(A,Xd,S)) ) ) ).

% DERIV_cmult_right
tff(fact_6668_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_nb(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3),topolo174197925503356063within(A,Xd,S)) ) ) ).

% DERIV_cmult
tff(fact_6669_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Xd: A,S: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,Xd,S)) ) ).

% DERIV_cmult_Id
tff(fact_6670_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xd: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xd,S))
         => has_field_derivative(A,aTP_Lamp_nc(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,Xd))),Db),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_field_derivative_cosh
tff(fact_6671_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xd: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xd,S))
         => has_field_derivative(A,aTP_Lamp_nd(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,Xd))),Db),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_field_derivative_sinh
tff(fact_6672_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xd,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ne(fun(A,A),fun(fun(A,A),fun(A,A)),F2),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,F2,Xd)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G,Xd))),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_mult'
tff(fact_6673_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,Xd: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,Xd,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xd,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ne(fun(A,A),fun(fun(A,A),fun(A,A)),F2),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,Xd))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,Xd))),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_mult
tff(fact_6674_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xd: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,S3))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),H6),S3)
                 => ( aa(real,$o,ord_less(real,H6),D2)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),H6))),aa(real,real,F2,Xd)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_6675_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xd: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,S3))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),H6),S3)
                 => ( aa(real,$o,ord_less(real,H6),D2)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,Xd)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xd),H6))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_6676_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F3: filter(A),G: fun(A,A),G2: A] :
          ( has_field_derivative(A,F2,F7,F3)
         => ( has_field_derivative(A,G,G2,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nf(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F7),G2),F3) ) ) ) ).

% field_differentiable_add
tff(fact_6677_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xd,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nf(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D3),E4),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_add
tff(fact_6678_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_ng(fun(A,A),fun(A,fun(A,A)),F2),C2),divide_divide(A,D3,C2),topolo174197925503356063within(A,Xd,S)) ) ) ).

% DERIV_cdivide
tff(fact_6679_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xd: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,S3))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,minus_minus(real,Xd),H6),S3)
                 => ( aa(real,$o,ord_less(real,H6),D2)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,minus_minus(real,Xd),H6))),aa(real,real,F2,Xd)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_6680_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xd: real,S3: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,S3))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,minus_minus(real,Xd),H6),S3)
                 => ( aa(real,$o,ord_less(real,H6),D2)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,Xd)),aa(real,real,F2,aa(real,real,minus_minus(real,Xd),H6))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_6681_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: filter(A)] : has_field_derivative(A,aTP_Lamp_nh(A,A),one_one(A),F3) ) ).

% DERIV_ident
tff(fact_6682_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F3: filter(A)] : has_field_derivative(A,aTP_Lamp_ni(A,fun(A,A),K),zero_zero(A),F3) ) ).

% DERIV_const
tff(fact_6683_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_nj(fun(A,A),fun(nat,fun(A,A)),F2),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),D3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xd)),Nb))),topolo174197925503356063within(A,Xd,S)) ) ) ).

% DERIV_power_Suc
tff(fact_6684_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X,top_top(set(real))))
       => ( aa(real,real,V,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2),aa(num,real,numeral_numeral(real),bit0(one2)))) = 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_6685_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_nk(fun(A,A),fun(nat,fun(A,A)),F2),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),D3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xd)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xd,S)) ) ) ).

% DERIV_power
tff(fact_6686_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xd: A,S: set(A)] :
          ( ( Xd != 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),Xd)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xd,S)) ) ) ).

% DERIV_inverse
tff(fact_6687_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,Xd: real,D: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
       => ( ! [Y2: real] :
              ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xd),Y2))),D)
             => aa(real,$o,ord_less_eq(real,aa(real,real,F2,Xd)),aa(real,real,F2,Y2)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_6688_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,Xd: real,D: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
       => ( ! [Y2: real] :
              ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xd),Y2))),D)
             => aa(real,$o,ord_less_eq(real,aa(real,real,F2,Y2)),aa(real,real,F2,Xd)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_6689_DERIV__ln__divide,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => has_field_derivative(real,ln_ln(real),divide_divide(real,one_one(real),Xd),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_6690_DERIV__pow,axiom,
    ! [Nb: nat,Xd: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_nl(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),Xd),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xd,S)) ).

% DERIV_pow
tff(fact_6691_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xd: A] :
          ( ! [Y2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),Y2))
         => has_field_derivative(A,aTP_Lamp_nm(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Xd)),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_6692_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,Xd: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),Xd)
         => ( aa(A,$o,ord_less(A,Xd),B2)
           => ( topolo174197925503356063within(A,Xd,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,Xd,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_6693_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,Xd: real,Nb: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_nn(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,Xd)),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))))),M),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_6694_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_6695_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,Xd: A,S: set(A),G: fun(A,A),E: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,Xd,S))
         => ( has_field_derivative(A,G,E,topolo174197925503356063within(A,Xd,S))
           => ( ( aa(A,A,G,Xd) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_mz(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,G,Xd))),aa(A,A,aa(A,fun(A,A),times_times(A),E),aa(A,A,F2,Xd))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,Xd)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xd,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_6696_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,Xd: A,S: set(A)] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,Xd,S))
         => ( ( aa(A,A,F2,Xd) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_my(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xd)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_6697_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [F4: fun(A,nat),N: nat] :
          ( ( image(A,nat,F4,A3) = collect(nat,aTP_Lamp_as(nat,fun(nat,$o),N)) )
          & inj_on(A,nat,F4,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6698_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ~ ! [F4: fun(A,nat)] :
            ( ? [N: nat] : image(A,nat,F4,A3) = collect(nat,aTP_Lamp_as(nat,fun(nat,$o),N))
           => ~ inj_on(A,nat,F4,A3) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_6699_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),F2: fun(A,A),F7: A,Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K6)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),Z2),aa(A,A,F2,Z2)) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K6)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Z),F7) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_6700_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_no(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,minus_minus(real,R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_6701_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xd: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_np(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
             => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,K6))
               => has_field_derivative(A,aTP_Lamp_nm(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Xd)),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_6702_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xd: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,K6))
           => has_field_derivative(A,aTP_Lamp_nm(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Xd)),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_6703_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K6)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)) )
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K6)
           => has_field_derivative(A,aTP_Lamp_nm(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_6704_summable__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X))
         => summable(real,comp(nat,real,nat,F2,G)) ) ) ) ).

% summable_reindex
tff(fact_6705_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,Xd: real,R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,G,Xd))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_nq(fun(real,real),fun(real,fun(real,real)),G),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G,Xd),aa(real,real,minus_minus(real,R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_6706_DERIV__log,axiom,
    ! [Xd: real,B2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => has_field_derivative(real,log(B2),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),Xd)),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_6707_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,Xd: real,F2: fun(real,real),R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,G,Xd))
       => ( has_field_derivative(real,F2,R2,topolo174197925503356063within(real,Xd,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_nr(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,Xd),aa(real,real,F2,Xd))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(real,real,ln_ln(real),aa(real,real,G,Xd)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F2,Xd)),aa(real,real,G,Xd)))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_6708_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_6709_DERIV__real__sqrt,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
     => has_field_derivative(real,sqrt,divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xd)),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_6710_DERIV__arctan,axiom,
    ! [Xd: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_6711_arsinh__real__has__field__derivative,axiom,
    ! [Xd: real,A3: set(real)] : has_field_derivative(real,arsinh(real),divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xd,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_6712_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( sin(A,Xd) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_6713_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xd: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G,Xd)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xd,S))
           => has_field_derivative(A,aTP_Lamp_ns(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tanh(A),aa(A,A,G,Xd))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_6714_DERIV__real__sqrt__generic,axiom,
    ! [Xd: real,D3: real] :
      ( ( Xd != zero_zero(real) )
     => ( ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
         => ( D3 = divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xd)),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,ord_less(real,Xd),zero_zero(real))
           => ( D3 = divide_divide(real,aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xd))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D3,topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_6715_arcosh__real__has__field__derivative,axiom,
    ! [Xd: real,A3: set(real)] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => has_field_derivative(real,arcosh(real),divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xd,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_6716_artanh__real__has__field__derivative,axiom,
    ! [Xd: real,A3: set(real)] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),one_one(real))
     => has_field_derivative(real,artanh(real),divide_divide(real,one_one(real),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xd,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_6717_suminf__reindex__mono,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X))
         => aa(real,$o,ord_less_eq(real,suminf(real,comp(nat,real,nat,F2,G))),suminf(real,F2)) ) ) ) ).

% suminf_reindex_mono
tff(fact_6718_DERIV__real__root,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xd)
       => has_field_derivative(real,root(Nb),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_6719_DERIV__arccos,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_6720_DERIV__arcsin,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_6721_suminf__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X))
         => ( ! [X: nat] :
                ( ~ member(nat,X,image(nat,nat,G,top_top(set(nat))))
               => ( aa(nat,real,F2,X) = zero_zero(real) ) )
           => ( suminf(real,comp(nat,real,nat,F2,G)) = suminf(real,F2) ) ) ) ) ) ).

% suminf_reindex
tff(fact_6722_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xd: real,Nb: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M2: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X),topolo174197925503356063within(real,X,top_top(set(real))))
       => ? [T6: real] :
            ( aa(real,$o,ord_less_eq(real,abs_abs(real,T6)),abs_abs(real,Xd))
            & ( aa(real,real,F2,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_nt(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_6723_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xd: real,Nb: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M2: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X),topolo174197925503356063within(real,X,top_top(set(real)))) )
     => ? [T6: real] :
          ( aa(real,$o,ord_less_eq(real,abs_abs(real,T6)),abs_abs(real,Xd))
          & ( aa(real,real,F2,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_nt(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_6724_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xd: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( ( Xd != 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(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xd)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_6725_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M2: nat,T6: real] :
                ( ( aa(nat,$o,ord_less(nat,M2),Nb)
                  & aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
                  & aa(real,$o,ord_less_eq(real,T6),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
           => ? [T6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),T6)
                & aa(real,$o,ord_less(real,T6),H)
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_nu(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_6726_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,ord_less(nat,M2),Nb)
                & aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
                & aa(real,$o,ord_less_eq(real,T6),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ? [T6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),T6)
              & aa(real,$o,ord_less_eq(real,T6),H)
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_nu(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_6727_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,H),zero_zero(real))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M2: nat,T6: real] :
                ( ( aa(nat,$o,ord_less(nat,M2),Nb)
                  & aa(real,$o,ord_less_eq(real,H),T6)
                  & aa(real,$o,ord_less_eq(real,T6),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
           => ? [T6: real] :
                ( aa(real,$o,ord_less(real,H),T6)
                & aa(real,$o,ord_less(real,T6),zero_zero(real))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_nu(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_6728_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xd: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( ( Xd != zero_zero(real) )
         => ( ! [M2: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X),topolo174197925503356063within(real,X,top_top(set(real))))
           => ? [T6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),abs_abs(real,T6))
                & aa(real,$o,ord_less(real,abs_abs(real,T6)),abs_abs(real,Xd))
                & ( aa(real,real,F2,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_nt(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_6729_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xd: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M2: nat,T6: real] :
            ( ( aa(nat,$o,ord_less(nat,M2),Nb)
              & aa(real,$o,ord_less_eq(real,abs_abs(real,T6)),abs_abs(real,Xd)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
       => ? [T6: real] :
            ( aa(real,$o,ord_less_eq(real,abs_abs(real,T6)),abs_abs(real,Xd))
            & ( aa(real,real,F2,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_nt(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xd),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_6730_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,ord_less(nat,M2),Nb)
                & aa(real,$o,ord_less_eq(real,A2),T6)
                & aa(real,$o,ord_less_eq(real,T6),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( aa(real,$o,ord_less(real,A2),C2)
           => ( aa(real,$o,ord_less_eq(real,C2),B2)
             => ? [T6: real] :
                  ( aa(real,$o,ord_less(real,A2),T6)
                  & aa(real,$o,ord_less(real,T6),C2)
                  & ( aa(real,real,F2,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_nv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,A2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_6731_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,ord_less(nat,M2),Nb)
                & aa(real,$o,ord_less_eq(real,A2),T6)
                & aa(real,$o,ord_less_eq(real,T6),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( aa(real,$o,ord_less_eq(real,A2),C2)
           => ( aa(real,$o,ord_less(real,C2),B2)
             => ? [T6: real] :
                  ( aa(real,$o,ord_less(real,C2),T6)
                  & aa(real,$o,ord_less(real,T6),B2)
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_nv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,B2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_6732_Taylor,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,Xd: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,ord_less(nat,M2),Nb)
                & aa(real,$o,ord_less_eq(real,A2),T6)
                & aa(real,$o,ord_less_eq(real,T6),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( aa(real,$o,ord_less_eq(real,A2),C2)
           => ( aa(real,$o,ord_less_eq(real,C2),B2)
             => ( aa(real,$o,ord_less_eq(real,A2),Xd)
               => ( aa(real,$o,ord_less_eq(real,Xd),B2)
                 => ( ( Xd != C2 )
                   => ? [T6: real] :
                        ( $ite(
                            aa(real,$o,ord_less(real,Xd),C2),
                            ( aa(real,$o,ord_less(real,Xd),T6)
                            & aa(real,$o,ord_less(real,T6),C2) ),
                            ( aa(real,$o,ord_less(real,C2),T6)
                            & aa(real,$o,ord_less(real,T6),Xd) ) )
                        & ( aa(real,real,F2,Xd) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_nw(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),Xd)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Xd),C2)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_6733_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => inj_on(real,real,aTP_Lamp_mk(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6734_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B4: real] :
      ( ! [M2: nat,T6: real] :
          ( ( aa(nat,$o,ord_less(nat,M2),Nb)
            & aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
            & aa(real,$o,ord_less_eq(real,T6),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
     => ( ( Nb = aa(nat,nat,suc,K) )
       => ! [M3: nat,T7: real] :
            ( ( aa(nat,$o,ord_less(nat,M3),Nb)
              & aa(real,$o,ord_less_eq(real,zero_zero(real)),T7)
              & aa(real,$o,ord_less_eq(real,T7),H) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_ny(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Nb),Diff),B4),M3),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_nz(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M3),T7)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,M3))))),aa(real,real,aa(real,fun(real,real),times_times(real),B4),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),T7),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,M3))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,M3))))))),topolo174197925503356063within(real,T7,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_6735_DERIV__arctan__series,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xd)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_oa(real,real),suminf(real,aTP_Lamp_ob(real,fun(nat,real),Xd)),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_6736_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X: real] :
          ( member(real,X,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_oc(fun(nat,real),fun(real,fun(nat,real)),F2),X)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_oe(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_oc(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_6737_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xd: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xd))
         => ( aa(real,$o,ord_less(real,aa(A,real,G,Xd)),one_one(real))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
             => has_derivative(A,real,aTP_Lamp_of(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_og(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xd),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_6738_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,ord_less(A,L),I)
            & aa(A,$o,ord_less(A,I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_6739_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_6740_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D3,F3)
        <=> has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D3),F3) ) ) ).

% has_field_derivative_def
tff(fact_6741_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: fun(A,A),F3: filter(A),D7: A] :
          ( has_derivative(A,A,F2,D3,F3)
         => ( ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),D7) = aa(A,A,D3,X)
           => has_field_derivative(A,F2,D7,F3) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_6742_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D3,F3)
         => has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D3),F3) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_6743_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F7: fun(A,real),Xd: A,S: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,Xd,S))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xd,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oh(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_oi(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xd),G),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_6744_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_6745_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F3: filter(A)] : has_derivative(A,B,aTP_Lamp_oj(B,fun(A,B),C2),aTP_Lamp_ok(A,B),F3) ) ).

% has_derivative_const
tff(fact_6746_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F3)
         => ( has_derivative(A,B,G,G2,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ol(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_ol(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G2),F3) ) ) ) ).

% has_derivative_add
tff(fact_6747_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G2: fun(A,B),F3: filter(A),Y: B] :
          ( has_derivative(A,B,G,G2,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_om(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_om(fun(A,B),fun(B,fun(A,B)),G2),Y),F3) ) ) ).

% has_derivative_mult_left
tff(fact_6748_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G2: fun(A,B),F3: filter(A),Xd: B] :
          ( has_derivative(A,B,G,G2,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_on(fun(A,B),fun(B,fun(A,B)),G),Xd),aa(B,fun(A,B),aTP_Lamp_on(fun(A,B),fun(B,fun(A,B)),G2),Xd),F3) ) ) ).

% has_derivative_mult_right
tff(fact_6749_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xd,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_op(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xd),G),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_mult
tff(fact_6750_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,B),Xd: A] :
          ( has_derivative(A,B,aTP_Lamp_ok(A,B),F3,topolo174197925503356063within(A,Xd,top_top(set(A))))
         => ! [X4: A] : aa(A,B,F3,X4) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_6751_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),Xd: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
         => has_derivative(A,real,aTP_Lamp_oq(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_or(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),Xd),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_derivative_exp
tff(fact_6752_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less_eq(A,B2),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_6753_tanh__real__bounds,axiom,
    ! [Xd: real] : member(real,aa(real,real,tanh(real),Xd),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) ).

% tanh_real_bounds
tff(fact_6754_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xd: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xd,S))
         => has_derivative(A,A,aTP_Lamp_nd(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,Xd))),Db)),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_derivative_sinh
tff(fact_6755_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xd: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xd,S))
         => has_derivative(A,A,aTP_Lamp_nc(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,Xd))),Db)),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_derivative_cosh
tff(fact_6756_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),Xd: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
         => has_derivative(A,real,aTP_Lamp_os(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ot(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),Xd),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_derivative_sin
tff(fact_6757_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less_eq(A,B2),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_6758_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less_eq(A,B2),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_6759_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S3: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S3))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xd,S3))
           => ( ( aa(A,B,G,Xd) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ou(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ov(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xd),G),G2),topolo174197925503356063within(A,Xd,S3)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_6760_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),Xd: B,F7: fun(B,A),S3: set(B)] :
          ( ( aa(B,A,F2,Xd) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F7,topolo174197925503356063within(B,Xd,S3))
           => has_derivative(B,A,aTP_Lamp_ow(fun(B,A),fun(B,A),F2),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_ox(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),Xd),F7),topolo174197925503356063within(B,Xd,S3)) ) ) ) ).

% has_derivative_inverse
tff(fact_6761_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: A,S3: set(A)] :
          ( ( Xd != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_oy(A,fun(A,A),Xd),topolo174197925503356063within(A,Xd,S3)) ) ) ).

% has_derivative_inverse'
tff(fact_6762_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F7: real,G: fun(A,real),Xd: A,G2: fun(A,real),S: set(A)] :
          ( has_field_derivative(real,F2,F7,topolo174197925503356063within(real,aa(A,real,G,Xd),top_top(set(real))))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_oz(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_pa(real,fun(fun(A,real),fun(A,real)),F7),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_6763_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),Xd: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
         => has_derivative(A,real,aTP_Lamp_pb(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_pc(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),Xd),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_derivative_cos
tff(fact_6764_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xd: real,Y: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( member(real,Xd,set_or5935395276787703475ssThan(real,A2,B2))
       => ( member(real,Y,set_or5935395276787703475ssThan(real,A2,B2))
         => ( ! [X: real] :
                ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) )
           => ( aa(real,real,F2,Xd) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_6765_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S3: set(A),Nb: nat] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S3))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_pd(fun(A,B),fun(nat,fun(A,B)),F2),Nb),aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_pe(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F7),Xd),Nb),topolo174197925503356063within(A,Xd,S3)) ) ) ).

% has_derivative_power
tff(fact_6766_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xd: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,Xd))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
           => has_derivative(A,real,aTP_Lamp_pf(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xd),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_ln
tff(fact_6767_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S3: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S3))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xd,S3))
           => ( ( aa(A,B,G,Xd) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ph(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pi(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xd),G),G2),topolo174197925503356063within(A,Xd,S3)) ) ) ) ) ).

% has_derivative_divide
tff(fact_6768_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I6: set(A),F2: fun(A,fun(B,C)),F7: fun(A,fun(B,C)),Xd: B,S3: set(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I3),aa(A,fun(B,C),F7,I3),topolo174197925503356063within(B,Xd,S3)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_pk(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I6),F2),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_pm(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I6),F2),F7),Xd),topolo174197925503356063within(B,Xd,S3)) ) ) ).

% has_derivative_prod
tff(fact_6769_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),Xd: A,X7: set(A),F2: fun(A,real),F7: fun(A,real)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,X7))
         => ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,Xd,X7))
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,Xd))
             => ( member(A,Xd,X7)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pn(fun(A,real),fun(fun(A,real),fun(A,real)),G),F2),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_po(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G2),Xd),F2),F7),topolo174197925503356063within(A,Xd,X7)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_6770_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xd: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,Xd))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
           => has_derivative(A,real,aTP_Lamp_pp(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xd),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_6771_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),Xd: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
         => has_derivative(A,real,aTP_Lamp_pr(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ps(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),Xd),topolo174197925503356063within(A,Xd,S)) ) ) ).

% has_derivative_arctan
tff(fact_6772_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xd: A,G2: fun(A,real),S: set(A)] :
          ( ( cos(real,aa(A,real,G,Xd)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
           => has_derivative(A,real,aTP_Lamp_pt(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pu(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xd),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_tan
tff(fact_6773_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F7: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L5: fun(nat,real)] :
      ( ! [N: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_pv(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N),aa(nat,real,aa(real,fun(nat,real),F7,X0),N),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X: real] :
            ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
           => summable(real,aa(real,fun(nat,real),F2,X)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,A2,B2))
         => ( summable(real,aa(real,fun(nat,real),F7,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X: real,Y2: real] :
                    ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
                   => ( member(real,Y2,set_or5935395276787703475ssThan(real,A2,B2))
                     => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),F2,X),N)),aa(nat,real,aa(real,fun(nat,real),F2,Y2),N)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N)),abs_abs(real,aa(real,real,minus_minus(real,X),Y2)))) ) )
               => has_field_derivative(real,aTP_Lamp_pw(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F7,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_6774_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xd: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xd))
         => ( aa(real,$o,ord_less(real,aa(A,real,G,Xd)),one_one(real))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xd,S))
             => has_derivative(A,real,aTP_Lamp_px(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_py(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xd),G2),topolo174197925503356063within(A,Xd,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_6775_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G: fun(B,real),Xd: B,F2: fun(real,A),G2: fun(B,real),S: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G,Xd),top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,aa(B,real,G,Xd)),ring_1_Ints(A))
           => ( has_derivative(B,real,G,G2,topolo174197925503356063within(B,Xd,S))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_pz(fun(B,real),fun(fun(real,A),fun(B,real)),G),F2),aTP_Lamp_qa(fun(B,real),fun(B,real),G2),topolo174197925503356063within(B,Xd,S)) ) ) ) ) ).

% has_derivative_floor
tff(fact_6776_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xd: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_np(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,K6))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qc(fun(nat,A),fun(A,fun(A,A)),C2),Xd),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_6777_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qd(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6778_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qe(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6779_power__tendsto__0__iff,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qf(nat,fun(fun(A,real),fun(A,real)),Nb),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% power_tendsto_0_iff
tff(fact_6780_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xd: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xd,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qg(A,fun(fun(A,B),fun(A,B)),Xd),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xd)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_6781_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qh(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6782_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qh(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6783_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qh(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6784_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qi(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6785_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),A2: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X: A] :
                ( ( X != A2 )
               => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,G,X)) )
           => ( ! [X: A] :
                  ( ( X != A2 )
                 => aa(real,$o,ord_less_eq(real,aa(A,real,G,X)),aa(A,real,F2,X)) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_6786_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A2: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_qj(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_6787_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),A2))),D5) )
                     => ( aa(A,B,F2,X) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_qk(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_6788_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S3))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ql(fun(A,A),fun(A,fun(A,A)),F2),Xd),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,Xd,S3)) ) ) ).

% has_field_derivative_iff
tff(fact_6789_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,S3))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ql(fun(A,A),fun(A,fun(A,A)),F2),Xd),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,Xd,S3)) ) ) ).

% has_field_derivativeD
tff(fact_6790_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_6791_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),A2)
       => ( aa(real,$o,ord_less(real,A2),one_one(real))
         => filterlim(A,real,aTP_Lamp_qm(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F3) ) ) ) ).

% tendsto_artanh
tff(fact_6792_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_qn(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Nb)),F3) ) ) ).

% tendsto_power
tff(fact_6793_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,nat,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,nat),fun(A,B),aTP_Lamp_qo(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_6794_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,nat),B2: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,B2),F3)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_qp(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),B2)),F3) ) ) ) ).

% tendsto_power_strong
tff(fact_6795_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),Nb: nat] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(nat,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% continuous_power
tff(fact_6796_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F3: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_qr(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_null_power
tff(fact_6797_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
         => ( ( A2 != one_one(real) )
           => ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qs(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F3) ) ) ) ) ) ).

% tendsto_log
tff(fact_6798_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qt(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_right_zero
tff(fact_6799_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qu(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_left_zero
tff(fact_6800_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_mult_zero
tff(fact_6801_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F3) ) ) ) ).

% tendsto_mult_one
tff(fact_6802_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qx(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C2)),F3) ) ) ).

% tendsto_mult_right
tff(fact_6803_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qy(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),L)),F3) ) ) ).

% tendsto_mult_left
tff(fact_6804_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),F3) ) ) ) ).

% tendsto_mult
tff(fact_6805_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_ra(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_6806_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_rb(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_6807_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_rc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_6808_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_rd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_6809_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_re(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_add_zero
tff(fact_6810_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_rf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_6811_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D: B,F3: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rg(B,fun(fun(A,B),fun(A,B)),C2),F2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),D)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D),F3) ) ) ).

% tendsto_add_const_iff
tff(fact_6812_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_re(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),F3) ) ) ) ).

% tendsto_add
tff(fact_6813_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),Xd: real,F3: filter(A),Nb: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xd),F3)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_rh(fun(A,real),fun(nat,fun(A,real)),F2),Nb),topolo7230453075368039082e_nhds(real,aa(real,real,root(Nb),Xd)),F3) ) ).

% tendsto_real_root
tff(fact_6814_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I6: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_ri(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,one_one(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_rj(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I6),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F3) ) ) ).

% tendsto_one_prod'
tff(fact_6815_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I6: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_rk(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_rl(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I6),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ).

% tendsto_null_sum
tff(fact_6816_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_rm(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sgn_sgn(B,L)),F3) ) ) ) ).

% tendsto_sgn
tff(fact_6817_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_rn(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_divide_zero
tff(fact_6818_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ro(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,divide_divide(B,A2,B2)),F3) ) ) ) ) ).

% tendsto_divide
tff(fact_6819_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_rp(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F3) ) ) ) ).

% tendsto_inverse
tff(fact_6820_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( cosh(B,A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_rq(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A2)),F3) ) ) ) ).

% tendsto_tanh
tff(fact_6821_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_rr(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F3) ) ) ) ).

% tendsto_tan
tff(fact_6822_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,aTP_Lamp_rs(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero
tff(fact_6823_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_rs(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_iff
tff(fact_6824_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_rs(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_cancel
tff(fact_6825_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform_eq
tff(fact_6826_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ru(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_cancel
tff(fact_6827_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform2
tff(fact_6828_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A2: B,F3: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rv(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform
tff(fact_6829_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ru(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_iff
tff(fact_6830_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ru(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% LIM_zero
tff(fact_6831_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( A2 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr
tff(fact_6832_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( ( A2 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_gc(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A2)),F3) ) ) ).

% tendsto_ln
tff(fact_6833_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),Xd: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xd),F3)
     => filterlim(A,real,aTP_Lamp_rx(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,Xd)),F3) ) ).

% tendsto_real_sqrt
tff(fact_6834_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,ord_less(real,one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_ry(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ).

% tendsto_arcosh
tff(fact_6835_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,real,aTP_Lamp_rz(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_norm_zero
tff(fact_6836_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_rz(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_iff
tff(fact_6837_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_rz(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_6838_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_sa(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F3) ) ) ) ).

% tendsto_cot
tff(fact_6839_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => ? [S2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),S2)
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A2))),S2) )
                   => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X4)),L5))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_6840_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
             => ? [S5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S5)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),A2))),S5) )
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X)),L5))),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6841_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S6)
                  & ! [X5: A] :
                      ( ( ( X5 != A2 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X5),A2))),S6) )
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X5)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_6842_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R)
         => ( ! [X: A] :
                ( ( X != A2 )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),A2))),R)
                 => ( aa(A,B,F2,X) = aa(A,B,G,X) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_6843_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D3: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sb(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sc(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6844_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [R3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,C2),X4))),R3) )
               => aa(real,$o,ord_less(real,aa(real,real,F2,X4)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_6845_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,C2),X4))),R3) )
               => ( aa(real,real,F2,X4) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_6846_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [R3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,C2),X4))),R3) )
               => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,F2,X4)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_6847_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),A2))),D5) )
                     => ( aa(A,B,F2,X) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_qk(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_6848_isCont__real__sqrt,axiom,
    ! [Xd: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_6849_isCont__real__root,axiom,
    ! [Xd: real,Nb: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),root(Nb)) ).

% isCont_real_root
tff(fact_6850_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(fun(A,B),fun(A,B),aTP_Lamp_sd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_6851_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_rf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_6852_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_rd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_6853_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),Nb: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% isCont_power
tff(fact_6854_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_se(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_6855_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_sf(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_6856_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sg(fun(A,A),fun(A,fun(A,A)),F2),Xd),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_6857_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xd,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sg(fun(A,A),fun(A,fun(A,A)),F2),Xd),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_6858_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_sh(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6859_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_6860_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,M: nat] : set_or5935395276787703475ssThan(nat,Nb,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),M)) ).

% greaterThanLessThan_upt
tff(fact_6861_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K6: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K)
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,H3)),K)
                 => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K6),real_V7770717601297561774m_norm(A,H3))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_6862_isCont__inverse__function2,axiom,
    ! [A2: real,Xd: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),Xd)
     => ( aa(real,$o,ord_less(real,Xd),B2)
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,A2),Z2)
             => ( aa(real,$o,ord_less_eq(real,Z2),B2)
               => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,ord_less_eq(real,A2),Z2)
               => ( aa(real,$o,ord_less_eq(real,Z2),B2)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xd),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_6863_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xd: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D3),topolo174197925503356063within(A,Xd,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sg(fun(A,A),fun(A,fun(A,A)),F2),Xd),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_6864_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_sd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_6865_isCont__ln,axiom,
    ! [Xd: real] :
      ( ( Xd != zero_zero(real) )
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),ln_ln(real)) ) ).

% isCont_ln
tff(fact_6866_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_sf(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_6867_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_si(fun(A,B),fun(A,fun(A,B)),F2),A2),F3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6868_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,S),F2)
         => ( ( cos(A,aa(A,A,F2,Xd)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,S),aTP_Lamp_rr(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_6869_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_6870_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,S),F2)
         => ( ( sin(A,aa(A,A,F2,Xd)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,S),aTP_Lamp_sa(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_6871_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xd: A,A3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xd,A3),F2)
         => ( ( cosh(B,aa(A,B,F2,Xd)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xd,A3),aTP_Lamp_sj(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_6872_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,Xd: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,Xd,top_top(set(A))))
        <=> ? [G3: fun(A,A)] :
              ( ! [Z5: A] : aa(A,A,minus_minus(A,aa(A,A,F2,Z5)),aa(A,A,F2,Xd)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Z5)),aa(A,A,minus_minus(A,Z5),Xd))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,top_top(set(A))),G3)
              & ( aa(A,A,G3,Xd) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_6873_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,ord_less_eq(real,A2),B2)
         => ( ! [X: real] :
                ( ( aa(real,$o,ord_less_eq(real,A2),X)
                  & aa(real,$o,ord_less_eq(real,X),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X4: real] :
                    ( ( aa(real,$o,ord_less_eq(real,A2),X4)
                      & aa(real,$o,ord_less_eq(real,X4),B2) )
                   => aa(A,$o,ord_less_eq(A,aa(real,A,F2,X4)),M8) )
                & ! [N11: A] :
                    ( aa(A,$o,ord_less(A,N11),M8)
                   => ? [X: real] :
                        ( aa(real,$o,ord_less_eq(real,A2),X)
                        & aa(real,$o,ord_less_eq(real,X),B2)
                        & aa(A,$o,ord_less(A,N11),aa(real,A,F2,X)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_6874_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A,D: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D)),F3,topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),D),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_6875_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D: A,F3: filter(B),A2: A] :
          ( filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D)),F3,topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),D),top_top(set(A))))
        <=> filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_6876_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( cos(A,Xd) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_6877_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( sin(A,Xd) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_6878_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( ( cosh(A,Xd) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_6879_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),S)
         => ( ! [X: A] :
                ( ( X != zero_zero(A) )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),S)
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),A2),X),aa(A,A,F2,X)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_6880_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),S)
         => ( ! [X: A] :
                ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),S)
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),A2),X),aa(A,A,F2,X)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_6881_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H3: A,N: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,H3)),K)
                   => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H3),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N)),real_V7770717601297561774m_norm(A,H3))) ) )
             => filterlim(A,B,aTP_Lamp_sk(fun(A,fun(nat,B)),fun(A,B),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_6882_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_rr(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_6883_isCont__arcosh,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_6884_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_sl(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_6885_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_sa(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_6886_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D3: real,G: fun(real,real),Xd: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D3,topolo174197925503356063within(real,aa(real,real,G,Xd),top_top(set(real))))
     => ( ( D3 != zero_zero(real) )
       => ( aa(real,$o,ord_less(real,A2),Xd)
         => ( aa(real,$o,ord_less(real,Xd),B2)
           => ( ! [Y2: real] :
                  ( aa(real,$o,ord_less(real,A2),Y2)
                 => ( aa(real,$o,ord_less(real,Y2),B2)
                   => ( aa(real,real,F2,aa(real,real,G,Y2)) = Y2 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D3),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_6887_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C2: fun(nat,A),Nb: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_sm(fun(nat,A),fun(nat,fun(A,A)),C2),Nb)) ) ).

% isCont_polynom
tff(fact_6888_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xd: A] :
          ( ! [Y2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),Y2))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,top_top(set(A))),aTP_Lamp_nm(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_6889_isCont__arccos,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_6890_isCont__arcsin,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_6891_LIM__less__bound,axiom,
    ! [B2: real,Xd: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,B2),Xd)
     => ( ! [X: real] :
            ( member(real,X,set_or5935395276787703475ssThan(real,B2,Xd))
           => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,F2,X)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),F2)
         => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,F2,Xd)) ) ) ) ).

% LIM_less_bound
tff(fact_6892_isCont__artanh,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xd,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_6893_isCont__inverse__function,axiom,
    ! [D: real,Xd: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),D)
     => ( ! [Z2: real] :
            ( aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,Z2),Xd))),D)
           => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,Z2),Xd))),D)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xd),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_6894_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G2: fun(real,real),F7: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ! [Z2: real] :
            ( aa(real,$o,ord_less_eq(real,A2),Z2)
           => ( aa(real,$o,ord_less_eq(real,Z2),B2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,A2),Z2)
             => ( aa(real,$o,ord_less_eq(real,Z2),B2)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,ord_less(real,A2),Z2)
               => ( aa(real,$o,ord_less(real,Z2),B2)
                 => has_field_derivative(real,G,aa(real,real,G2,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( aa(real,$o,ord_less(real,A2),Z2)
                 => ( aa(real,$o,ord_less(real,Z2),B2)
                   => has_field_derivative(real,F2,aa(real,real,F7,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C4: real] :
                  ( aa(real,$o,ord_less(real,A2),C4)
                  & aa(real,$o,ord_less(real,C4),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G2,C4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F7,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_6895_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Xd: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,Xd,top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,Xd),ring_1_Ints(A))
           => has_field_derivative(real,aTP_Lamp_sn(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_6896_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xd: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),real_V7770717601297561774m_norm(A,K6))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xd,top_top(set(A))),aTP_Lamp_nm(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_6897_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),C2: fun(nat,B),K6: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_so(fun(nat,B),fun(B,fun(nat,B)),C2),K6))
           => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,A2))),real_V7770717601297561774m_norm(B,K6))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_sq(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_6898_summable__Leibniz_I2_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(nat,real,A2,zero_zero(nat)))
         => ! [N9: nat] : member(real,suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6899_summable__Leibniz_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( aa(real,$o,ord_less(real,aa(nat,real,A2,zero_zero(nat))),zero_zero(real))
         => ! [N9: nat] : member(real,suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_6900_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ss(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_6901_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_st(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_6902_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_su(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_6903_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),A2: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sv(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_6904_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),K: nat,A2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sv(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_6905_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),Nb: nat] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aa(nat,fun(A,real),aTP_Lamp_sw(fun(A,real),fun(nat,fun(A,real)),F2),Nb)) ) ) ).

% continuous_real_root
tff(fact_6906_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_sx(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_6907_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,Xd: A] :
          ( aa(A,$o,ord_less(A,Y),Xd)
         => ? [U2: fun(nat,A)] :
              ( ! [N9: nat] : aa(A,$o,ord_less(A,aa(nat,A,U2,N9)),Xd)
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,Xd),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_6908_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ? [U2: fun(nat,A)] :
              ( ! [N9: nat] : aa(A,$o,ord_less(A,Xd),aa(nat,A,U2,N9))
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,Xd),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_6909_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_sy(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_6910_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_sy(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_6911_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_6912_continuous__at__within__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),G)
           => ( ( aa(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_6913_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xd: A,S: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xd,S),F2)
         => ( ( aa(A,real,F2,Xd) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xd,S),aTP_Lamp_ta(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_6914_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => filterlim(nat,nat,aTP_Lamp_tb(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6915_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_6916_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_tc(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_6917_isCont__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_6918_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xd: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xd,top_top(set(A))),F2)
         => ( ( aa(A,real,F2,Xd) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xd,top_top(set(A))),aTP_Lamp_ta(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_6919_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_td(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6920_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X7: fun(nat,A),Xd: A,L: nat] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,Xd),at_top(nat))
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_te(fun(nat,A),fun(nat,fun(nat,A)),X7),L),topolo7230453075368039082e_nhds(A,Xd),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6921_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_tf(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_6922_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_tg(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_6923_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_th(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_6924_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,G,aa(nat,nat,suc,N))),aa(nat,real,G,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),aa(nat,real,G,N))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ti(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L3: real] :
                ( ! [N9: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N9)),L3)
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L3),at_top(nat))
                & ! [N9: nat] : aa(real,$o,ord_less_eq(real,L3),aa(nat,real,G,N9))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_6925_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N11: nat] :
        ! [N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N11),N)
         => aa(real,$o,ord_less(real,R3),aa(nat,real,X7,N)) )
     => filterlim(nat,real,aTP_Lamp_tj(fun(nat,real),fun(nat,real),X7),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6926_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_tk(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_6927_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_tl(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_6928_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
     => filterlim(nat,real,aTP_Lamp_tm(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6929_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_tn(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_6930_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_to(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_6931_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),L)
       => ( ! [E2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
             => ? [N9: nat] : aa(real,$o,ord_less_eq(real,L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N9)),E2)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6932_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_tp(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6933_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_tq(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6934_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_tr(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6935_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_th(fun(nat,A),fun(nat,A),F2),aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_6936_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_tg(fun(nat,A),fun(nat,A),F2),aa(A,A,minus_minus(A,C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_6937_LIMSEQ__realpow__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less(real,Xd),one_one(real))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),Xd),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_6938_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xd: real,A2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_ts(real,fun(real,fun(nat,real)),Xd),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6939_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,C2)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_6940_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,C2)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_6941_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xd)
     => filterlim(nat,real,aTP_Lamp_tt(real,fun(nat,real),Xd),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6942_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,F2,S)
        <=> filterlim(nat,A,aTP_Lamp_tu(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_6943_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_tv(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_6944_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),Xd: real] :
          ( filterlim(nat,real,aTP_Lamp_tw(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Xd),at_top(nat))
         => ( aa(real,$o,ord_less(real,Xd),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_6945_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_to(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_6946_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [No: nat] :
                ! [N6: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No),N6)
                 => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X7,N6)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_6947_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No2),N)
                 => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X7,N)),L5))),R3) ) )
         => filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6948_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => ? [No3: nat] :
              ! [N9: nat] :
                ( aa(nat,$o,ord_less_eq(nat,No3),N9)
               => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X7,N9)),L5))),R2) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6949_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xd: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),one_one(real))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xd),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_6950_tendsto__exp__limit__sequentially,axiom,
    ! [Xd: real] : filterlim(nat,real,aTP_Lamp_tx(real,fun(nat,real),Xd),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xd)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_6951_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F3: filter(A),Xd: B] :
          ( filterlim(A,nat,F2,at_top(nat),F3)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,Xd)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ty(fun(A,nat),fun(B,fun(A,B)),F2),Xd),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_power_zero
tff(fact_6952_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_tz(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_6953_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_6954_summable__Leibniz_I1_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => summable(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_6955_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : aa(nat,A,S,N) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ua(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_6956_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xd: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xd)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_ub(A,fun(nat,A),Xd),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6957_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xd: A] :
          ( aa(real,$o,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,Xd))
         => filterlim(nat,A,aTP_Lamp_uc(A,fun(nat,A),Xd),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6958_summable,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => summable(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_6959_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_ud(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K2: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ue(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_6960_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_uf(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K2: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ue(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_6961_summable__Leibniz_I4_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_ug(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_6962_zeroseq__arctan__series,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xd)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_av(real,fun(nat,real),Xd),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_6963_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6964_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_ug(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6965_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L3: real] :
              ( ! [N9: nat] : aa(real,$o,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)))),L3)
              & filterlim(nat,real,aTP_Lamp_ug(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat))
              & ! [N9: nat] : aa(real,$o,ord_less_eq(real,L3),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_uh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6966_summable__Leibniz_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_uh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6967_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_uh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6968_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,ord_less_eq(real,suminf(real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_6969_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_ll(fun(nat,A),fun(nat,A),F2),F3,at_top(nat))
    <=> filterlim(nat,A,F2,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6970_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ui(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),Xd),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xd,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6971_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xd: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),Xd)) ) ).

% bounded_linear_mult_right
tff(fact_6972_bounded__linear__mult__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Y: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_om(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_6973_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Xd: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_on(fun(A,B),fun(B,fun(A,B)),G),Xd)) ) ) ).

% bounded_linear_const_mult
tff(fact_6974_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_uj(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_6975_real__bounded__linear,axiom,
    ! [F2: fun(real,real)] :
      ( real_V3181309239436604168linear(real,real,F2)
    <=> ? [C5: real] :
        ! [X5: real] : aa(real,real,F2,X5) = aa(real,real,aa(real,fun(real,real),times_times(real),X5),C5) ) ).

% real_bounded_linear
tff(fact_6976_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ol(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_6977_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_uk(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_6978_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_ok(A,B)) ) ).

% bounded_linear_zero
tff(fact_6979_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
            ! [X4: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K8)) ) ) ).

% bounded_linear.bounded
tff(fact_6980_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ul(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_6981_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,ord_less_eq(real,zero_zero(real)),K8)
              & ! [X4: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K8)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_6982_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K8)
              & ! [X4: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_6983_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K6: real] :
          ( ! [X: A,Y2: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F2,Y2))
         => ( ! [R3: real,X: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R3),X)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F2,X))
           => ( ! [X: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K6))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6984_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6985_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_um(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F7),Xd),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_6986_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_un(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),Xd),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_6987_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F7: fun(A,B),Xd: A,F2: fun(A,B),S: set(A)] :
          ( real_V3181309239436604168linear(A,B,F7)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_uo(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F7),Xd),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xd,S))
           => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivativeI
tff(fact_6988_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & ? [E3: fun(A,B)] :
                ( ! [H2: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),H2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xd)),aa(A,B,F7,H2))),aa(A,B,E3,H2))
                & filterlim(A,real,aTP_Lamp_up(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_6989_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ui(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),Xd),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% has_derivative_within
tff(fact_6990_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D3: fun(A,B),Xd: A] :
          ( has_derivative(A,B,F2,D3,topolo174197925503356063within(A,Xd,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D3)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_uq(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D3),Xd),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6991_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F7,F3)
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_us(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F7),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% has_derivative_def
tff(fact_6992_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xd: A,S3: set(A),F2: fun(A,B),F7: fun(A,B)] :
          ( member(A,Xd,S3)
         => ( topolo1002775350975398744n_open(A,S3)
           => ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S3))
            <=> ( real_V3181309239436604168linear(A,B,F7)
                & ? [E3: fun(A,B)] :
                    ( ! [H2: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),H2),S3)
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),H2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xd)),aa(A,B,F7,H2))),aa(A,B,E3,H2)) ) )
                    & filterlim(A,real,aTP_Lamp_up(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_6993_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),Xd: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( member(A,Xd,S3)
           => ( aa(A,$o,ord_less(A,Xd),Y)
             => ? [B3: A] :
                  ( aa(A,$o,ord_less(A,Xd),B3)
                  & aa(set(A),$o,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,Xd,B3)),S3) ) ) ) ) ) ).

% open_right
tff(fact_6994_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_sd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6995_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_se(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_6996_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_sf(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_6997_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_6998_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ta(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_6999_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_uu(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_rr(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_7000_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_uu(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_sa(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_7001_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_sj(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_7002_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( aa(real,$o,ord_less(real,one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_uv(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_7003_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))))
                 => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_to(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_7004_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( member(real,aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ut(A,A))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_uw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_7005_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S3: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( member(B,A2,S3)
           => ( topolo1002775350975398744n_open(B,S3)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S3))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ux(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_7006_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E: real,F7: fun(A,B),S: set(A),Xd: A,F2: fun(A,B),H7: fun(A,real)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),E)
         => ( real_V3181309239436604168linear(A,B,F7)
           => ( ! [Y2: A] :
                  ( member(A,Y2,S)
                 => ( ( Y2 != Xd )
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y2,Xd)),E)
                     => aa(real,$o,ord_less_eq(real,divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,F2,Y2)),aa(A,B,F2,Xd))),aa(A,B,F7,aa(A,A,minus_minus(A,Y2),Xd)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y2),Xd)))),aa(A,real,H7,Y2)) ) ) )
             => ( filterlim(A,real,H7,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xd,S))
               => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_7007_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A] : real_V557655796197034286t_dist(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)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel
tff(fact_7008_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [B2: A,A2: A,C2: A] : real_V557655796197034286t_dist(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)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel2
tff(fact_7009_dist__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,Y: A] :
          ( ( real_V557655796197034286t_dist(A,Xd,Y) = zero_zero(real) )
        <=> ( Xd = Y ) ) ) ).

% dist_eq_0_iff
tff(fact_7010_dist__self,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A] : real_V557655796197034286t_dist(A,Xd,Xd) = zero_zero(real) ) ).

% dist_self
tff(fact_7011_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] : real_V557655796197034286t_dist(A,zero_zero(A),Xd) = real_V7770717601297561774m_norm(A,Xd) ) ).

% dist_0_norm
tff(fact_7012_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,Y: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,Xd,Y))
        <=> ( Xd != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_7013_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,Y: A] :
          ( aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(A,Xd,Y)),zero_zero(real))
        <=> ( Xd = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_7014_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: real,A2: A,Y: real] : real_V557655796197034286t_dist(A,aa(A,A,real_V8093663219630862766scaleR(A,Xd),A2),aa(A,A,real_V8093663219630862766scaleR(A,Y),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),abs_abs(real,aa(real,real,minus_minus(real,Xd),Y))),real_V7770717601297561774m_norm(A,A2)) ) ).

% dist_scaleR
tff(fact_7015_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,D: real] : topolo1002775350975398744n_open(A,collect(A,aa(real,fun(A,$o),aTP_Lamp_uy(A,fun(real,fun(A,$o)),Xd),D))) ) ).

% open_ball
tff(fact_7016_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,Z: A,Y: A,E: real] :
          ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xd,Z)),real_V557655796197034286t_dist(A,Y,Z))),E)
         => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Xd,Y)),E) ) ) ).

% dist_triangle_lt
tff(fact_7017_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E1: real,X2: A,E22: real] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X15,Y)),E1)
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X2,Y)),E22)
           => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X15,X2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_7018_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,Xd: A,E: real] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y,Xd)),E)
         => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Xd,Y)),E) ) ) ).

% dist_commute_lessI
tff(fact_7019_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,Y: A] : aa(real,$o,ord_less_eq(real,zero_zero(real)),real_V557655796197034286t_dist(A,Xd,Y)) ) ).

% zero_le_dist
tff(fact_7020_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,Y: A] : ~ aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Xd,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_7021_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xd: A,Y: A] :
          ( ( Xd != Y )
         => aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,Xd,Y)) ) ) ).

% dist_pos_lt
tff(fact_7022_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xd: A] : real_V7770717601297561774m_norm(A,Xd) = real_V557655796197034286t_dist(A,Xd,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_7023_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S3: set(A)] :
          ( topolo1002775350975398744n_open(A,S3)
        <=> ! [X5: A] :
              ( member(A,X5,S3)
             => ? [E3: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
                  & ! [Y5: A] :
                      ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y5,X5)),E3)
                     => member(A,Y5,S3) ) ) ) ) ) ).

% open_dist
tff(fact_7024_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G4: filter(B),Xd: A,S3: set(A),F3: filter(B),D: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G4,topolo174197925503356063within(A,Xd,S3))
         => ( aa(filter(B),$o,ord_less_eq(filter(B),G4),F3)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
             => ( ! [X3: A] :
                    ( member(A,X3,S3)
                   => ( aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X3,Xd))
                     => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X3,Xd)),D)
                       => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) ) )
               => filterlim(A,B,F2,F3,topolo174197925503356063within(A,Xd,S3)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_7025_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,A2: A,S3: set(A),D: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,A2,S3))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
           => ( member(A,A2,S3)
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,A2)),D)
                     => ( aa(A,A,F2,X) = aa(A,A,G,X) ) ) )
               => has_field_derivative(A,G,F7,topolo174197925503356063within(A,A2,S3)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_7026_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xd: A,S: set(A),D: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xd,S))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
           => ( member(A,Xd,S)
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X3,Xd)),D)
                     => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
               => has_derivative(A,B,G,F7,topolo174197925503356063within(A,Xd,S)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_7027_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
             => ? [M10: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M10),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M10),N)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,M2),aa(nat,A,X7,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% metric_CauchyI
tff(fact_7028_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X7)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E)
           => ? [M8: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M8),M3)
               => ! [N9: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,M8),N9)
                   => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,M3),aa(nat,A,X7,N9))),E) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_7029_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [N7: nat] :
                ! [N6: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,N7),N6)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,S,N6),aa(nat,A,S,N7))),E3) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_7030_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),M4)
                 => ! [N6: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M9),N6)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,M4),aa(nat,A,X7,N6))),E3) ) ) ) ) ) ).

% Cauchy_def
tff(fact_7031_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xd,Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_7032_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xd,Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,minus_minus(A,Xd),Y)) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_7033_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xd: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xd,Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_7034_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xd: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xd,Y),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_7035_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xd: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xd,Y),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,minus_minus(B,B2),C2)) = aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_7036_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xd: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xd,Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,minus_minus(B,A2),B2)),C2) = aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_7037_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E: real,X2: A] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X15,Y)),divide_divide(real,E,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X2,Y)),divide_divide(real,E,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X15,X2)),E) ) ) ) ).

% dist_triangle_half_l
tff(fact_7038_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X15: A,E: real,X2: A] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y,X15)),divide_divide(real,E,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y,X2)),divide_divide(real,E,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X15,X2)),E) ) ) ) ).

% dist_triangle_half_r
tff(fact_7039_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,X2: A,E: real,X32: A,X42: A] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X15,X2)),divide_divide(real,E,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X2,X32)),divide_divide(real,E,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
           => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X32,X42)),divide_divide(real,E,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
             => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X15,X42)),E) ) ) ) ) ).

% dist_triangle_third
tff(fact_7040_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,Xd: A,S3: set(A),D: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xd,S3))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
           => ( ! [X3: A] :
                  ( member(A,X3,S3)
                 => ( aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X3,Xd))
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X3,Xd)),D)
                     => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xd,S3)) ) ) ) ) ).

% Lim_transform_within
tff(fact_7041_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),M4)
                 => ! [N6: nat] :
                      ( aa(nat,$o,ord_less(nat,M4),N6)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,F2,M4),aa(nat,A,F2,N6))),E3) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_7042_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
             => ? [M10: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M10),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,ord_less(nat,M2),N)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,M2),aa(nat,A,X7,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% CauchyI'
tff(fact_7043_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_uz(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_dist_iff
tff(fact_7044_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S6)
                  & ! [X5: A] :
                      ( ( ( X5 != A2 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X5,A2)),S6) )
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X5),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_7045_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => ? [S2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),S2)
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X4,A2)),S2) )
                   => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L5)),R2) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_7046_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
             => ? [S5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S5)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,A2)),S5) )
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X),L5)),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_7047_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R)
           => ( ! [X: A] :
                  ( ( X != A2 )
                 => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,A2)),R)
                   => ( aa(A,B,F2,X) = aa(A,B,G,X) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_7048_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [No: nat] :
                ! [N6: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No),N6)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,N6),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_7049_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No2),N)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,N),L5)),R3) ) )
         => filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_7050_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => ? [No3: nat] :
              ! [N9: nat] :
                ( aa(nat,$o,ord_less_eq(nat,No3),N9)
               => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,N9),L5)),R2) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_7051_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: A,Nb: nat] :
          ( nO_MATCH(A,A,one_one(A),Xd)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xd)),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),Xd),Nb)) ) ) ) ).

% power_minus'
tff(fact_7052_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [J2: nat] :
            ? [M9: nat] :
            ! [M4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M9),M4)
             => ! [N6: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),N6)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,M4),aa(nat,A,X7,N6))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J2)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_7053_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xd,Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xd) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xd)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_7054_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,A2)),D5) )
                     => ( aa(A,B,F2,X) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_va(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_7055_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xd: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xd,Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),Xd) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xd)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xd)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_7056_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,A2)),D5) )
                     => ( aa(A,B,F2,X) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_va(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_7057_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [No: nat] :
                  ( aa(nat,$o,ord_less(nat,zero_zero(nat)),No)
                  & ! [N6: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,No),N6)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X7,N6),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_7058_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ux(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_7059_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xd: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xd,B2)
         => ( ( B2 != zero_zero(B) )
           => ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_7060_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xd: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xd,B2)
         => ( ( B2 != zero_zero(B) )
           => ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_7061_less__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F5: filter(A)] :
      ( aa(filter(A),$o,ord_less(filter(A),F3),F5)
    <=> ( aa(filter(A),$o,ord_less_eq(filter(A),F3),F5)
        & ~ aa(filter(A),$o,ord_less_eq(filter(A),F5),F3) ) ) ).

% less_filter_def
tff(fact_7062_tendsto__exp__limit__at__right,axiom,
    ! [Xd: real] : filterlim(real,real,aTP_Lamp_vb(real,fun(real,real),Xd),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xd)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_7063_tendsto__arctan__at__bot,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_7064_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_greaterThan(A,K))
        <=> aa(A,$o,ord_less(A,K),I) ) ) ).

% greaterThan_iff
tff(fact_7065_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_greaterThan(A,L) = collect(A,ord_less(A,L)) ) ).

% greaterThan_def
tff(fact_7066_ln__at__0,axiom,
    filterlim(real,real,ln_ln(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% ln_at_0
tff(fact_7067_artanh__real__at__right__1,axiom,
    filterlim(real,real,artanh(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),one_one(real)),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_7068_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( topolo174197925503356063within(A,A2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_7069_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_vc(fun(real,A),fun(real,fun(real,A)),F2),A2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_7070_filterlim__tan__at__right,axiom,
    filterlim(real,real,tan(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))))) ).

% filterlim_tan_at_right
tff(fact_7071_exp__at__bot,axiom,
    filterlim(real,real,exp(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_bot(real)) ).

% exp_at_bot
tff(fact_7072_filterlim__inverse__at__bot__neg,axiom,
    filterlim(real,real,inverse_inverse(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_lessThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_bot_neg
tff(fact_7073_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P3: B,F1: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P3,set_ord_greaterThan(B,P3)),F1)
         => ( aa(B,$o,ord_less(B,zero_zero(B)),C2)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C2),P3) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vd(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,set_ord_greaterThan(B,L)),F1) ) ) ) ) ).

% filterlim_times_pos
tff(fact_7074_log__inj,axiom,
    ! [B2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => inj_on(real,real,log(B2),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_7075_tanh__real__at__bot,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),one_one(real))),at_bot(real)) ).

% tanh_real_at_bot
tff(fact_7076_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ve(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_7077_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),set_ord_greaterThan(real,one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_7078_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X: real] :
          ( aa(real,$o,ord_less_eq(real,X),B2)
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
              & aa(real,$o,ord_less(real,zero_zero(real)),Y3) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,ord_less(real,Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_7079_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vf(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_bot(real),F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_7080_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vf(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_top(real),F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_7081_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_vg(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_7082_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ve(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_7083_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_7084_greaterThan__0,axiom,
    set_ord_greaterThan(nat,zero_zero(nat)) = image(nat,nat,suc,top_top(set(nat))) ).

% greaterThan_0
tff(fact_7085_filterlim__pow__at__top,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qf(nat,fun(fun(A,real),fun(A,real)),Nb),F2),at_top(real),F3) ) ) ).

% filterlim_pow_at_top
tff(fact_7086_greaterThan__Suc,axiom,
    ! [K: nat] : set_ord_greaterThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_greaterThan(nat,K)),insert(nat,aa(nat,nat,suc,K),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_7087_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vh(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_7088_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vh(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_7089_tanh__real__at__top,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(real)) ).

% tanh_real_at_top
tff(fact_7090_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vi(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% real_tendsto_divide_at_top
tff(fact_7091_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),set_ord_lessThan(real,one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_7092_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => filterlim(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_inverse_0_at_top
tff(fact_7093_tendsto__inverse__0,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_infinity(A)) ) ).

% tendsto_inverse_0
tff(fact_7094_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ve(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_7095_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vk(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_7096_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F2: fun(A,real),F3: filter(A)] :
      ( aa(real,$o,ord_less(real,S),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vl(real,fun(fun(A,real),fun(A,real)),S),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_neg_powr
tff(fact_7097_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F3)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_7098_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_divide_0
tff(fact_7099_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_vo(fun(A,B),fun(nat,fun(A,B)),F2),Nb),at_infinity(B),F3) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_7100_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_vp(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_7101_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_vq(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_7102_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_vq(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_7103_filterlim__inverse__at__top__right,axiom,
    filterlim(real,real,inverse_inverse(real),at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_top_right
tff(fact_7104_filterlim__inverse__at__right__top,axiom,
    filterlim(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))),at_top(real)) ).

% filterlim_inverse_at_right_top
tff(fact_7105_filterlim__inverse__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),at_infinity(A),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% filterlim_inverse_at_infinity
tff(fact_7106_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ve(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_7107_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_vr(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_7108_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_rp(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F3)
        <=> filterlim(A,B,G,at_infinity(B),F3) ) ) ).

% filterlim_inverse_at_iff
tff(fact_7109_tendsto__exp__limit__at__top,axiom,
    ! [Xd: real] : filterlim(real,real,aTP_Lamp_vs(real,fun(real,real),Xd),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xd)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_7110_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F3: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F3)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_mz(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_7111_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))),set_ord_lessThan(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% filterlim_tan_at_left
tff(fact_7112_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_7113_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X: real] :
          ( aa(real,$o,ord_less_eq(real,B2),X)
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
              & aa(real,$o,ord_less(real,Y3),zero_zero(real)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,ord_less(real,Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_7114_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xd: A] :
          ( aa(real,$o,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,Xd))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xd),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_7115_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,comp(A,A,A,F2,inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_7116_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,Nb: nat,B4: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),K)
           => ( aa(nat,$o,ord_less_eq(nat,K),Nb)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_vt(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Nb),B4),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_7117_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),Xd: real,G2: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
     => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_7118_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_ml(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_7119_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_vx(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_7120_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xd: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xd,set_ord_greaterThan(A,Xd)))
        <=> ? [B7: A] :
              ( aa(A,$o,ord_less(A,Xd),B7)
              & ! [Y5: A] :
                  ( aa(A,$o,ord_less(A,Xd),Y5)
                 => ( aa(A,$o,ord_less(A,Y5),B7)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_7121_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xd: A,Y: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( eventually(A,P,topolo174197925503356063within(A,Xd,set_ord_greaterThan(A,Xd)))
          <=> ? [B7: A] :
                ( aa(A,$o,ord_less(A,Xd),B7)
                & ! [Y5: A] :
                    ( aa(A,$o,ord_less(A,Xd),Y5)
                   => ( aa(A,$o,ord_less(A,Y5),B7)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_7122_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,ord_less(A,C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_7123_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N7: A] :
            ! [N6: A] :
              ( aa(A,$o,ord_less(A,N7),N6)
             => aa(A,$o,P,N6) ) ) ) ).

% eventually_at_top_dense
tff(fact_7124_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xd: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xd,set_ord_lessThan(A,Xd)))
        <=> ? [B7: A] :
              ( aa(A,$o,ord_less(A,B7),Xd)
              & ! [Y5: A] :
                  ( aa(A,$o,ord_less(A,B7),Y5)
                 => ( aa(A,$o,ord_less(A,Y5),Xd)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_7125_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xd: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,Y),Xd)
         => ( eventually(A,P,topolo174197925503356063within(A,Xd,set_ord_lessThan(A,Xd)))
          <=> ? [B7: A] :
                ( aa(A,$o,ord_less(A,B7),Xd)
                & ! [Y5: A] :
                    ( aa(A,$o,ord_less(A,B7),Y5)
                   => ( aa(A,$o,ord_less(A,Y5),Xd)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_7126_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_vx(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_7127_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_vy(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_7128_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N7: A] :
            ! [N6: A] :
              ( aa(A,$o,ord_less(A,N6),N7)
             => aa(A,$o,P,N6) ) ) ) ).

% eventually_at_bot_dense
tff(fact_7129_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,B2),top_top(A))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B7: A] :
                ( aa(A,$o,ord_less(A,B7),top_top(A))
                & ! [Z5: A] :
                    ( aa(A,$o,ord_less(A,B7),Z5)
                   => aa(A,$o,P,Z5) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_7130_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,ord_less(B,Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3) ) ) ) ).

% order_tendstoD(2)
tff(fact_7131_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,ord_less(B,A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3) ) ) ) ).

% order_tendstoD(1)
tff(fact_7132_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [A5: A] :
              ( aa(A,$o,ord_less(A,A5),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_wb(fun(B,A),fun(A,fun(B,$o)),F2),A5),F3) )
         => ( ! [A5: A] :
                ( aa(A,$o,ord_less(A,Y),A5)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_wc(fun(B,A),fun(A,fun(B,$o)),F2),A5),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3) ) ) ) ).

% order_tendstoI
tff(fact_7133_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Xd: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xd),F3)
        <=> ( ! [L4: B] :
                ( aa(B,$o,ord_less(B,L4),Xd)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,$o)),F2),L4),F3) )
            & ! [U3: B] :
                ( aa(B,$o,ord_less(B,Xd),U3)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,$o)),F2),U3),F3) ) ) ) ) ).

% order_tendsto_iff
tff(fact_7134_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_top_dense
tff(fact_7135_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xd: A] : eventually(A,ord_less(A,Xd),topolo174197925503356063within(A,Xd,set_ord_greaterThan(A,Xd))) ) ).

% eventually_at_right_less
tff(fact_7136_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_we(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_bot_dense
tff(fact_7137_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A,S3: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S3))
        <=> ? [D4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),D4)
              & ! [X5: A] :
                  ( member(A,X5,S3)
                 => ( ( ( X5 != A2 )
                      & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X5,A2)),D4) )
                   => aa(A,$o,P,X5) ) ) ) ) ) ).

% eventually_at
tff(fact_7138_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),D4)
              & ! [X5: A] :
                  ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X5,A2)),D4)
                 => aa(A,$o,P,X5) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_7139_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X: A] :
              ( member(A,X,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X) )
         => ( aa(A,$o,ord_less(A,A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_7140_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X: A] :
              ( member(A,X,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X) )
         => ( aa(A,$o,ord_less(A,A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_7141_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_wf(fun(A,$o),fun(A,fun(A,$o)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_7142_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F3: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_wg(B,fun(fun(A,B),fun(A,$o)),L),F2),F3)
         => ( ! [X: B] :
                ( aa(B,$o,ord_less(B,L),X)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,$o)),F2),X),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% decreasing_tendsto
tff(fact_7143_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_wh(fun(A,B),fun(B,fun(A,$o)),F2),L),F3)
         => ( ! [X: B] :
                ( aa(B,$o,ord_less(B,X),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,$o)),F2),X),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% increasing_tendsto
tff(fact_7144_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less(B,C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wi(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7145_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less(B,Z7),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wj(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_7146_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_wk(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,F2,at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_7147_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_wl(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,F2,at_bot(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_7148_eventually__at__right__to__0,axiom,
    ! [P: fun(real,$o),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
    <=> eventually(real,aa(real,fun(real,$o),aTP_Lamp_wm(fun(real,$o),fun(real,fun(real,$o)),P),A2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_right_to_0
tff(fact_7149_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( eventually(A,aTP_Lamp_wn(fun(A,real),fun(A,$o),F2),F3)
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_uv(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_7150_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_wo(real,fun(real,fun(real,$o)),A2),B2),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ).

% eventually_at_right_real
tff(fact_7151_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( aa(real,$o,ord_less(real,B2),A2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_wo(real,fun(real,fun(real,$o)),B2),A2),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ).

% eventually_at_left_real
tff(fact_7152_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A,S3: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S3))
        <=> ? [D4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),D4)
              & ! [X5: A] :
                  ( member(A,X5,S3)
                 => ( ( ( X5 != A2 )
                      & aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(A,X5,A2)),D4) )
                   => aa(A,$o,P,X5) ) ) ) ) ) ).

% eventually_at_le
tff(fact_7153_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P3: fun(A,$o)] :
          ( eventually(A,P3,at_infinity(A))
        <=> ? [B7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),B7)
              & ! [X5: A] :
                  ( aa(real,$o,ord_less_eq(real,B7),real_V7770717601297561774m_norm(A,X5))
                 => aa(A,$o,P3,X5) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_7154_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,$o)),F2),L5),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_7155_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,$o)),F2),L5),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_greaterThan(B,L5)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_7156_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),E: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_wp(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E),F3) ) ) ) ).

% tendstoD
tff(fact_7157_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( ! [E2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_wp(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2),F3) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% tendstoI
tff(fact_7158_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_wp(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E3),F3) ) ) ) ).

% tendsto_iff
tff(fact_7159_eventually__at__top__to__right,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_wq(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_7160_eventually__at__right__to__top,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> eventually(real,aTP_Lamp_wq(fun(real,$o),fun(real,$o),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_7161_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),A2)
       => ( eventually(A,aTP_Lamp_wr(fun(A,real),fun(A,$o),F2),F3)
         => filterlim(A,real,aTP_Lamp_ry(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_7162_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,Q,X)
             => ( aa(A,$o,Q,Y2)
               => ( aa(A,$o,ord_less_eq(A,X),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,F2,Y2)) ) ) )
         => ( ! [X: B] :
                ( aa(B,$o,P,X)
               => ( aa(A,B,F2,aa(B,A,G,X)) = X ) )
           => ( ! [X: B] :
                  ( aa(B,$o,P,X)
                 => aa(A,$o,Q,aa(B,A,G,X)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,ord_less(A,B3),A2) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_7163_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X: A,Y2: A] :
              ( aa(A,$o,Q,X)
             => ( aa(A,$o,Q,Y2)
               => ( aa(A,$o,ord_less_eq(A,X),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,F2,Y2)) ) ) )
         => ( ! [X: B] :
                ( aa(B,$o,P,X)
               => ( aa(A,B,F2,aa(B,A,G,X)) = X ) )
           => ( ! [X: B] :
                  ( aa(B,$o,P,X)
                 => aa(A,$o,Q,aa(B,A,G,X)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,ord_less(A,A2),B3) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_7164_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,C),K6: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_ws(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K6),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% tendsto_0_le
tff(fact_7165_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wt(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_floor_less
tff(fact_7166_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,F2: fun(A,B),F3: filter(A)] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),C2)
         => ( filterlim(A,B,F2,at_infinity(B),F3)
          <=> ! [R5: real] :
                ( aa(real,$o,ord_less(real,C2),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_wu(fun(A,B),fun(real,fun(A,$o)),F2),R5),F3) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_7167_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wv(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_less_ceiling
tff(fact_7168_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( ( A2 != zero_zero(real) )
            | ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
              & eventually(A,aTP_Lamp_ww(fun(A,real),fun(A,$o),F2),F3) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr'
tff(fact_7169_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_ww(fun(A,real),fun(A,$o),F2),F3)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ) ).

% tendsto_powr2
tff(fact_7170_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_ww(fun(A,real),fun(A,$o),F2),F3)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_7171_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => ( eventually(A,aTP_Lamp_wk(fun(A,real),fun(A,$o),G),F3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vi(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_7172_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_wk(fun(A,real),fun(A,$o),F2),F3)
     => ( filterlim(A,real,F2,at_top(real),F3)
      <=> filterlim(A,real,comp(real,real,A,inverse_inverse(real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_7173_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_wk(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_inverse_at_top
tff(fact_7174_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_wk(fun(A,real),fun(A,$o),F2),F3)
     => ( filterlim(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),F2),at_top(real),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_7175_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_wl(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ) ).

% filterlim_inverse_at_bot
tff(fact_7176_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_7177_lhopital,axiom,
    ! [F2: fun(real,real),Xd: real,G: fun(real,real),G2: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xd,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xd,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,Xd,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Xd,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Xd,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F3,topolo174197925503356063within(real,Xd,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_7178_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_7179_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_7180_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_7181_lhopital__at__top,axiom,
    ! [G: fun(real,real),Xd: real,G2: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xd,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,Xd,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Xd,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Xd,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xd,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xd,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_7182_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G2: fun(real,real),F2: fun(real,real),F7: fun(real,real),Xd: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),topolo7230453075368039082e_nhds(real,Xd),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Xd),at_top(real)) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_7183_lhopital__right,axiom,
    ! [F2: fun(real,real),Xd: real,G: fun(real,real),G2: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
       => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
         => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F3,topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_7184_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G2: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( filterlim(real,real,G0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G0),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F0),F7),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G0),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_7185_lhopital__left,axiom,
    ! [F2: fun(real,real),Xd: real,G: fun(real,real),G2: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
       => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
         => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F3,topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,Xd,set_ord_lessThan(real,Xd))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_7186_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_7187_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_7188_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),Xd: real,G2: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
     => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xd,set_ord_greaterThan(real,Xd))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_7189_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G2: fun(real,real),F2: fun(real,real),F7: fun(real,real),Xd: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( eventually(real,aTP_Lamp_vu(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),topolo7230453075368039082e_nhds(real,Xd),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Xd),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_7190_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_wy(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_7191_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [Y5: B,K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_wz(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y5),K7),F3) ) ) ) ).

% Bfun_metric_def
tff(fact_7192_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_7193_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_xa(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_7194_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),K: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xb(fun(nat,A),fun(nat,fun(nat,A)),X7),K),at_top(nat))
         => bfun(nat,A,X7,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_7195_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_xa(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_7196_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),K: nat] :
          ( bfun(nat,A,X7,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xb(fun(nat,A),fun(nat,fun(nat,A)),X7),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_7197_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( bfun(nat,A,F2,at_top(nat))
         => ( bfun(nat,A,G,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_7198_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cr(A,fun(fun(nat,A),fun(nat,A)),C2),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_7199_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [N6: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X7,N6))),K7) ) ) ) ).

% Bseq_def
tff(fact_7200_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K6: real,X7: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
         => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X7,N))),K6)
           => bfun(nat,A,X7,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_7201_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
         => ~ ! [K8: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),K8)
               => ~ ! [N9: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X7,N9))),K8) ) ) ) ).

% BseqE
tff(fact_7202_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
         => ? [K8: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K8)
              & ! [N9: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X7,N9))),K8) ) ) ) ).

% BseqD
tff(fact_7203_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [N7: nat] :
            ! [N6: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(nat,A,X7,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% Bseq_iff1a
tff(fact_7204_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [N7: nat] :
            ! [N6: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X7,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% Bseq_iff
tff(fact_7205_Bseq__realpow,axiom,
    ! [Xd: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xd)
     => ( aa(real,$o,ord_less_eq(real,Xd),one_one(real))
       => bfun(nat,real,aa(real,fun(nat,real),power_power(real),Xd),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_7206_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K3)
              & ? [X5: A] :
                ! [N6: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X7,N6)),aa(A,A,uminus_uminus(A),X5)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_7207_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K3)
              & ? [N7: nat] :
                ! [N6: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X7,N6)),aa(A,A,uminus_uminus(A),aa(nat,A,X7,N7))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_7208_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( A2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_rp(fun(A,B),fun(A,B),F2),F3) ) ) ) ).

% Bfun_inverse
tff(fact_7209_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
         => ~ ! [B9: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),B9)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_xd(fun(A,B),fun(real,fun(A,$o)),F2),B9),F3) ) ) ) ).

% BfunE
tff(fact_7210_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_xd(fun(A,B),fun(real,fun(A,$o)),F2),K7),F3) ) ) ) ).

% Bfun_def
tff(fact_7211_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_xe(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_7212_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( ! [F4: fun(nat,A)] :
                ( ! [N9: nat] : aa(A,$o,ord_less(A,A2),aa(nat,A,F4,N9))
               => ( ! [N9: nat] : aa(A,$o,ord_less(A,aa(nat,A,F4,N9)),B2)
                 => ( order_antimono(nat,A,F4)
                   => ( filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_xf(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_7213_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,ord_less(A,L),I)
            & aa(A,$o,ord_less_eq(A,I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_7214_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,ord_less(A,K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_7215_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less(A,K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_7216_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_7217_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,A2: A,B2: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),C2),set_or3652927894154168847AtMost(A,A2,B2)) = set_or3652927894154168847AtMost(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)) ) ).

% image_add_greaterThanAtMost
tff(fact_7218_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A3)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,A3,aa(nat,nat,suc,I))),aa(nat,A,A3,I)) ) ) ).

% decseq_SucD
tff(fact_7219_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X7,aa(nat,nat,suc,N))),aa(nat,A,X7,N))
         => order_antimono(nat,A,X7) ) ) ).

% decseq_SucI
tff(fact_7220_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N6: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,F2,aa(nat,nat,suc,N6))),aa(nat,A,F2,N6)) ) ) ).

% decseq_Suc_iff
tff(fact_7221_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_7222_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2)) ) ) ).

% infinite_Ioc
tff(fact_7223_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),Xd: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( member(A,Xd,S3)
           => ( aa(A,$o,ord_less(A,Y),Xd)
             => ? [B3: A] :
                  ( aa(A,$o,ord_less(A,B3),Xd)
                  & aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,B3,Xd)),S3) ) ) ) ) ) ).

% open_left
tff(fact_7224_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,M: nat] : set_or3652927894154168847AtMost(nat,Nb,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_7225_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_or3652927894154168847AtMost(nat,M,Nb))) ) ) ) ).

% sum.head
tff(fact_7226_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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_or3652927894154168847AtMost(nat,M,Nb))) ) ) ) ).

% prod.head
tff(fact_7227_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less_eq(A,B2),D) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_7228_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less(A,B2),D) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_7229_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less_eq(A,C2),A2)
              & aa(A,$o,ord_less_eq(A,B2),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_7230_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,X7: fun(A,B),L5: B] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( ! [S7: fun(nat,A)] :
                ( ! [N9: nat] : aa(A,$o,ord_less(A,A2),aa(nat,A,S7,N9))
               => ( ! [N9: nat] : aa(A,$o,ord_less(A,aa(nat,A,S7,N9)),B2)
                 => ( order_antimono(nat,A,S7)
                   => ( filterlim(nat,A,S7,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_xg(fun(A,B),fun(fun(nat,A),fun(nat,B)),X7),S7),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X7,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_7231_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F3: filter(A)] :
          ( topolo6773858410816713723filter(A,F3)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [P7: fun(A,$o)] :
                  ( eventually(A,P7,F3)
                  & ! [X5: A,Y5: A] :
                      ( ( aa(A,$o,P7,X5)
                        & aa(A,$o,P7,Y5) )
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X5,Y5)),E3) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_7232_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_xh(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_inverse_code
tff(fact_7233_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_7234_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(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_7235_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_7236_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(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_7237_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),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_7238_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_7239_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2) )
     => aa(int,$o,ord_less(int,zero_zero(int)),Q2) ) ).

% quotient_of_denom_pos
tff(fact_7240_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_7241_word__atLeastAtMost__Suc__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A),U: word(A)] :
          ( ( M != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_or3652927894154168847AtMost(word(A),M,U) = set_or1337092689740270186AtMost(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),M),one_one(word(A))),U) ) ) ) ).

% word_atLeastAtMost_Suc_greaterThanAtMost
tff(fact_7242_word__range__minus__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( A2 != zero_zero(word(A)) )
         => ( set_or3652927894154168847AtMost(word(A),aa(word(A),word(A),minus_minus(word(A),A2),one_one(word(A))),B2) = set_or1337092689740270186AtMost(word(A),A2,B2) ) ) ) ).

% word_range_minus_1'
tff(fact_7243_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(of_int(A2)) = aa(int,product_prod(int,int),product_Pair(int,int,A2),one_one(int)) ).

% quotient_of_int
tff(fact_7244_GMVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ! [X: real] :
            ( ( aa(real,$o,ord_less_eq(real,A2),X)
              & aa(real,$o,ord_less_eq(real,X),B2) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F2) )
       => ( ! [X: real] :
              ( ( aa(real,$o,ord_less(real,A2),X)
                & aa(real,$o,ord_less(real,X),B2) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X,top_top(set(real)))) )
         => ( ! [X: real] :
                ( ( aa(real,$o,ord_less_eq(real,A2),X)
                  & aa(real,$o,ord_less_eq(real,X),B2) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),G) )
           => ( ! [X: real] :
                  ( ( aa(real,$o,ord_less(real,A2),X)
                    & aa(real,$o,ord_less(real,X),B2) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & aa(real,$o,ord_less(real,A2),C4)
                  & aa(real,$o,ord_less(real,C4),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B2)),aa(real,real,G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_7245_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q2: fun(A,B),C2: B,T: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_xi(fun(A,B),fun(B,fun(A,B)),Q2),C2),topolo174197925503356063within(A,T,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,T,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_7246_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C2: B,Q2: fun(A,B),T: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xj(B,fun(fun(A,B),fun(A,B)),C2),Q2),topolo174197925503356063within(A,T,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,T,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_7247_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),Xd: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xd,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xd,S))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% differentiable_mult
tff(fact_7248_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F3)
         => ( differentiable(A,B,G,F3)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ol(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F3) ) ) ) ).

% differentiable_add
tff(fact_7249_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xd: A,S: set(A),Nb: nat] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xd,S))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_pd(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo174197925503356063within(A,Xd,S)) ) ) ).

% differentiable_power
tff(fact_7250_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa(rat,$o,ord_less(rat,zero_zero(rat)),R2)
     => ~ ! [S2: rat] :
            ( aa(rat,$o,ord_less(rat,zero_zero(rat)),S2)
           => ! [T6: rat] :
                ( aa(rat,$o,ord_less(rat,zero_zero(rat)),T6)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S2),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_7251_abs__rat__def,axiom,
    ! [A2: rat] :
      abs_abs(rat,A2) = $ite(aa(rat,$o,ord_less(rat,A2),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A2),A2) ).

% abs_rat_def
tff(fact_7252_sgn__rat__def,axiom,
    ! [A2: 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_7253_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xd: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xd,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xd,S))
           => ( ( aa(A,B,G,Xd) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ou(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xd,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_7254_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xd: A,S: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xd,S))
         => ( ( aa(A,B,F2,Xd) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_xk(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,Xd,S)) ) ) ) ).

% differentiable_inverse
tff(fact_7255_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),aa(num,int,numeral_numeral(int),K))) = divide_divide(rat,one_one(rat),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_7256_normalize__negative,axiom,
    ! [Q2: int,P3: int] :
      ( aa(int,$o,ord_less(int,Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_7257_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),zero_zero(int))) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_7258_less__eq__rat__def,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa(rat,$o,ord_less_eq(rat,Xd),Y)
    <=> ( aa(rat,$o,ord_less(rat,Xd),Y)
        | ( Xd = Y ) ) ) ).

% less_eq_rat_def
tff(fact_7259_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2) )
     => aa(int,$o,ord_less(int,zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_7260_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P3: int,R2: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,R2),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_7261_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,A2),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_7262_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),A2)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_7263_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_7264_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),product_Pair(int,int,aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = divide_divide(rat,aa(num,rat,numeral_numeral(rat),K),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_7265_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(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_7266_MVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X: real] :
              ( aa(real,$o,ord_less(real,A2),X)
             => ( aa(real,$o,ord_less(real,X),B2)
               => differentiable(real,real,F2,topolo174197925503356063within(real,X,top_top(set(real)))) ) )
         => ? [L3: real,Z2: real] :
              ( aa(real,$o,ord_less(real,A2),Z2)
              & aa(real,$o,ord_less(real,Z2),B2)
              & has_field_derivative(real,F2,L3,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),L3) ) ) ) ) ) ).

% MVT
tff(fact_7267_set__bits__int__def,axiom,
    ! [F2: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,F2) = $ite(
        ? [N6: nat] :
        ! [M4: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N6),M4)
         => ( aa(nat,$o,F2,M4) = aa(nat,$o,F2,N6) ) ),
        $let(
          n2: nat,
          n2:= ord_Least(nat,aTP_Lamp_xl(fun(nat,$o),fun(nat,$o),F2)),
          aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list(nat),list($o),map(nat,$o,F2),upt(zero_zero(nat),aa(nat,nat,suc,n2))))) ),
        zero_zero(int) ) ).

% set_bits_int_def
tff(fact_7268_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_7269_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(A,B,G,X) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_xm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_7270_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(A,B,F2,X) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_xn(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_7271_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_xo(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_7272_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_xp(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_7273_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,Xd: A,B2: A,F2: fun(A,B)] :
          ( aa(A,$o,ord_less(A,A2),Xd)
         => ( aa(A,$o,ord_less(A,Xd),B2)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( aa(B,$o,ord_less(B,aa(A,B,F2,A2)),aa(A,B,F2,Xd))
                    & aa(B,$o,ord_less(B,aa(A,B,F2,Xd)),aa(A,B,F2,B2)) )
                  | ( aa(B,$o,ord_less(B,aa(A,B,F2,B2)),aa(A,B,F2,Xd))
                    & aa(B,$o,ord_less(B,aa(A,B,F2,Xd)),aa(A,B,F2,A2)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_7274_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_xq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_add
tff(fact_7275_continuous__on__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),Nb: nat] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aa(nat,fun(A,B),aTP_Lamp_xr(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% continuous_on_power
tff(fact_7276_continuous__on__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( topolo81223032696312382ous_on(A,nat,A3,G)
           => topolo81223032696312382ous_on(A,B,A3,aa(fun(A,nat),fun(A,B),aTP_Lamp_xs(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_power'
tff(fact_7277_continuous__on__mult__const,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [S: set(A),C2: A] : topolo81223032696312382ous_on(A,A,S,aa(A,fun(A,A),times_times(A),C2)) ) ).

% continuous_on_mult_const
tff(fact_7278_continuous__on__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aa(B,fun(A,B),aTP_Lamp_xt(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_right
tff(fact_7279_continuous__on__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aa(B,fun(A,B),aTP_Lamp_xu(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_left
tff(fact_7280_continuous__on__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( topolo81223032696312382ous_on(A,B,A3,G)
           => topolo81223032696312382ous_on(A,B,A3,aa(fun(A,B),fun(A,B),aTP_Lamp_xv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult'
tff(fact_7281_continuous__on__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_xw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult
tff(fact_7282_continuous__on__real__root,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),Nb: nat] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => topolo81223032696312382ous_on(A,real,S,aa(nat,fun(A,real),aTP_Lamp_xx(fun(A,real),fun(nat,fun(A,real)),F2),Nb)) ) ) ).

% continuous_on_real_root
tff(fact_7283_continuous__on__real__sqrt,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_xy(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_real_sqrt
tff(fact_7284_Least__Suc,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ml(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_7285_continuous__on__powr,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(A,real,F2,X) != zero_zero(real) ) )
             => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_xz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr
tff(fact_7286_continuous__on__ln,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(A,real,F2,X) != zero_zero(real) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_ya(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_ln
tff(fact_7287_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,$o,P,zero_zero(nat))
         => ( ! [K2: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K2))
              <=> aa(nat,$o,Q,K2) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7288_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(A,B,F2,X) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_yb(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_7289_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_yc(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_7290_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( cos(A,aa(A,A,F2,X)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_rr(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_7291_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( sin(A,aa(A,A,F2,X)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_sa(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_7292_continuous__on__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => ( cosh(B,aa(A,B,F2,X)) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,A3,aTP_Lamp_yd(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_7293_continuous__on__arcosh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X: real] :
            ( member(real,X,A3)
           => aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,F2,X)) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_ye(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_7294_continuous__on__powr_H,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,F2,X))
                    & ( ( aa(A,real,F2,X) = zero_zero(real) )
                     => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,X)) ) ) )
             => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_xz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_7295_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,X)) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ( aa(A,real,F2,X) != one_one(real) ) )
               => ( ! [X: A] :
                      ( member(A,X,S)
                     => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,X)) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_yf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_7296_continuous__on__arccos_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arccos) ).

% continuous_on_arccos'
tff(fact_7297_continuous__on__arcsin_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arcsin) ).

% continuous_on_arcsin'
tff(fact_7298_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X))
                  & aa(real,$o,ord_less_eq(real,aa(A,real,F2,X)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_yg(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_7299_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X))
                  & aa(real,$o,ord_less_eq(real,aa(A,real,F2,X)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_yh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_7300_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_yi(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_7301_continuous__on__artanh,axiom,
    ! [A3: set(real)] :
      ( aa(set(real),$o,ord_less_eq(set(real),A3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
     => topolo81223032696312382ous_on(real,real,A3,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_7302_continuous__on__artanh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X: real] :
            ( member(real,X,A3)
           => member(real,aa(real,real,F2,X),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_yj(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_artanh'
tff(fact_7303_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F7: fun(real,fun(real,real))] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X: real] :
                ( aa(real,$o,ord_less(real,A2),X)
               => ( aa(real,$o,ord_less(real,X),B2)
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F7,X),topolo174197925503356063within(real,X,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,ord_less(real,A2),Z2)
                & aa(real,$o,ord_less(real,Z2),B2)
                & ! [X4: real] : aa(real,real,aa(real,fun(real,real),F7,Z2),X4) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_7304_mvt,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F7: fun(real,fun(real,real))] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X: real] :
              ( aa(real,$o,ord_less(real,A2),X)
             => ( aa(real,$o,ord_less(real,X),B2)
               => has_derivative(real,real,F2,aa(real,fun(real,real),F7,X),topolo174197925503356063within(real,X,top_top(set(real)))) ) )
         => ~ ! [Xi3: real] :
                ( aa(real,$o,ord_less(real,A2),Xi3)
               => ( aa(real,$o,ord_less(real,Xi3),B2)
                 => ( aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) != aa(real,real,aa(real,fun(real,real),F7,Xi3),aa(real,real,minus_minus(real,B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_7305_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,ord_less(A,A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_7306_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,ord_less(A,A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_7307_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,ord_less(real,A2),X)
           => ( aa(real,$o,ord_less(real,X),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,ord_less(real,zero_zero(real)),Y3) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => aa(real,$o,ord_less(real,aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_7308_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,ord_less(real,A2),X)
           => ( aa(real,$o,ord_less(real,X),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,ord_less(real,Y3),zero_zero(real)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => aa(real,$o,ord_less(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_7309_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X: real] :
              ( aa(real,$o,ord_less(real,A2),X)
             => ( aa(real,$o,ord_less(real,X),B2)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B2) = aa(real,real,F2,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_7310_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),Xd: real] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X: real] :
              ( aa(real,$o,ord_less(real,A2),X)
             => ( aa(real,$o,ord_less(real,X),B2)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) ) )
         => ( aa(real,$o,ord_less_eq(real,A2),Xd)
           => ( aa(real,$o,ord_less_eq(real,Xd),B2)
             => ( aa(real,real,F2,Xd) = aa(real,real,F2,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_7311_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)))
           => ( ! [X: A] :
                  ( aa(A,$o,ord_less(A,A2),X)
                 => ( aa(A,$o,ord_less(A,X),B2)
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) )
             => ( aa(A,$o,ord_less(A,A2),B2)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_7312_Rolle,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X: real] :
                ( aa(real,$o,ord_less(real,A2),X)
               => ( aa(real,$o,ord_less(real,X),B2)
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,ord_less(real,A2),Z2)
                & aa(real,$o,ord_less(real,Z2),B2)
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_7313_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xd,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( (Y)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                          ( ! [X5: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => vEBT_VEBT_valid(X5,n2) )
                          & vEBT_VEBT_valid(Summary2,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X8)
                              & ! [X5: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ) ),
                              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,$o))))),aTP_Lamp_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_7314_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xd,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n2: nat,
                      n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $let(
                        m: nat,
                        m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                        ( ! [X5: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_VEBT_valid(X5,n2) )
                        & vEBT_VEBT_valid(Summary2,m)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X8)
                            & ! [X5: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ) ),
                            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,$o))))),aTP_Lamp_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_7315_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_yl(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_7316_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_ym(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_7317_option_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F12: A,F22: fun(B,A)] : case_option(A,B,F12,F22,none(B)) = F12 ).

% option.simps(4)
tff(fact_7318_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xd: option(A)] :
      ( case_option($o,A,(P),Q,Xd)
     => ( ( ( Xd = none(A) )
         => ~ (P) )
       => ~ ! [Y2: A] :
              ( ( Xd = aa(A,option(A),some(A),Y2) )
             => ~ aa(A,$o,Q,Y2) ) ) ) ).

% case_optionE
tff(fact_7319_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F12: A,F22: fun(B,A),Option: option(B)] :
      case_option(A,B,F12,F22,Option) = $ite(Option = none(B),F12,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ).

% option.case_eq_if
tff(fact_7320_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: option(A),Y: option(A)] :
          ( aa(option(A),$o,ord_less(option(A),Xd),Y)
        <=> case_option($o,A,$false,aTP_Lamp_yo(option(A),fun(A,$o),Xd),Y) ) ) ).

% less_option_def
tff(fact_7321_option_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F12: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F12,F22,Option))
    <=> ~ ( ( ( Option = none(B) )
            & ~ aa(A,$o,P,F12) )
          | ( ( 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_7322_option_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F12: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F12,F22,Option))
    <=> ( ( ( Option = none(B) )
         => aa(A,$o,P,F12) )
        & ( ( 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_7323_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_yp(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_7324_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg4: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary),Deg4)
    <=> ( ( Deg = Deg4 )
        & $let(
            n2: nat,
            n2:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m: nat,
              m:= aa(nat,nat,minus_minus(nat,Deg),n2),
              ( ! [X5: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                 => vEBT_VEBT_valid(X5,n2) )
              & vEBT_VEBT_valid(Summary,m)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                  & ! [X5: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ) ),
                  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,$o))))),aTP_Lamp_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n2),m)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_7325_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xd,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n2: nat,
                    n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                    $let(
                      m: nat,
                      m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                      ( ! [X5: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                         => vEBT_VEBT_valid(X5,n2) )
                      & vEBT_VEBT_valid(Summary2,m)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X8)
                          & ! [X5: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ) ),
                          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,$o))))),aTP_Lamp_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_7326_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xd,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( (Y)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                            ( ! [X5: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => vEBT_VEBT_valid(X5,n2) )
                            & vEBT_VEBT_valid(Summary2,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X8)
                                & ! [X5: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ) ),
                                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,$o))))),aTP_Lamp_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_7327_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xd,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(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,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                            ( ! [X5: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => vEBT_VEBT_valid(X5,n2) )
                            & vEBT_VEBT_valid(Summary2,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X8)
                                & ! [X5: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ) ),
                                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,$o))))),aTP_Lamp_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_7328_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) )
     => ( ! [X: A] :
            ( ( V = aa(A,option(A),some(A),X) )
           => hoare_hoare_triple(B,P,aa(A,heap_Time_Heap(B),Fs,X),Q) )
       => hoare_hoare_triple(B,P,case_option(heap_Time_Heap(B),A,Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_7329_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xd,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xd),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(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,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                          ( ! [X5: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => vEBT_VEBT_valid(X5,n2) )
                          & vEBT_VEBT_valid(Summary2,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X8)
                              & ! [X5: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ) ),
                              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,$o))))),aTP_Lamp_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_7330_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_yq(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_7331_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_yr(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_7332_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_7333_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_7334_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_7335_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_ys(num,option(num)),bit_take_bit_num(Nb,M)) ).

% take_bit_num_simps(3)
tff(fact_7336_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_7337_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_ys(num,option(num)),bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_7338_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_7339_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_7340_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_7341_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_7342_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_7343_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_7344_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_7345_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_7346_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_7347_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_7348_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_7349_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_7350_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_7351_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_7352_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_yt(num,option(num)),bit_and_not_num(M,Nb)) ).

% and_not_num.simps(8)
tff(fact_7353_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_7354_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_7355_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_7356_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_yu(num,fun(nat,option(num)),M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_7357_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_yv(num,fun(nat,option(num)),M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_7358_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num] : case_nat(A,A2,F2,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F2,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_7359_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num,Nb: nat] : case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),Nb)) ).

% case_nat_add_eq_if
tff(fact_7360_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F12: B,F22: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F12,F22,Nat)) = case_nat(A,aa(B,A,H,F12),aa(fun(nat,B),fun(nat,A),aTP_Lamp_yw(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_7361_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F12: A,F22: fun(nat,A)] : case_nat(A,F12,F22,zero_zero(nat)) = F12 ).

% old.nat.simps(4)
tff(fact_7362_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F12: A,F22: fun(nat,A),X2: nat] : case_nat(A,F12,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ).

% old.nat.simps(5)
tff(fact_7363_nth__Cons,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xd),Xs)),Nb) = case_nat(A,Xd,nth(A,Xs),Nb) ).

% nth_Cons
tff(fact_7364_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xd: A,F2: fun(nat,A),Nb: nat] :
      case_nat(A,Xd,F2,Nb) = $ite(Nb = zero_zero(nat),Xd,aa(nat,A,F2,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_7365_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_yx(nat,option(num)),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_7366_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_7367_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_7368_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_7369_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_mj(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_7370_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_mh(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_7371_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_7372_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xd: A] :
        ? [Y2: A] : aa(A,$o,ord_less(A,Y2),Xd) ) ).

% lt_ex
tff(fact_7373_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xd: A] :
        ? [X_1: A] : aa(A,$o,ord_less(A,Xd),X_1) ) ).

% gt_ex
tff(fact_7374_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ? [Z2: A] :
              ( aa(A,$o,ord_less(A,Xd),Z2)
              & aa(A,$o,ord_less(A,Z2),Y) ) ) ) ).

% dense
tff(fact_7375_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( Xd != Y ) ) ) ).

% less_imp_neq
tff(fact_7376_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_7377_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_7378_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_7379_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y3: A] :
                  ( aa(A,$o,ord_less(A,Y3),X)
                 => aa(A,$o,P,Y3) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_7380_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xd: A] :
          ( ~ aa(A,$o,ord_less(A,Y),Xd)
         => ( ~ aa(A,$o,ord_less(A,Xd),Y)
          <=> ( Xd = Y ) ) ) ) ).

% antisym_conv3
tff(fact_7381_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xd),Y)
         => ( ( Xd != Y )
           => aa(A,$o,ord_less(A,Y),Xd) ) ) ) ).

% linorder_cases
tff(fact_7382_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_7383_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,A2),A2) ) ).

% dual_order.irrefl
tff(fact_7384_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
        <=> ? [N6: A] :
              ( aa(A,$o,P,N6)
              & ! [M4: A] :
                  ( aa(A,$o,ord_less(A,M4),N6)
                 => ~ aa(A,$o,P,M4) ) ) ) ) ).

% exists_least_iff
tff(fact_7385_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A5: A,B3: A] :
              ( aa(A,$o,ord_less(A,A5),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A5),B3) )
         => ( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),P,A5),A5)
           => ( ! [A5: A,B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B3),A5)
                 => aa(A,$o,aa(A,fun(A,$o),P,A5),B3) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_7386_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_7387_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xd),Y)
        <=> ( aa(A,$o,ord_less(A,Y),Xd)
            | ( Xd = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_7388_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_7389_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_7390_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_7391_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( ( Xd != Y )
         => ( ~ aa(A,$o,ord_less(A,Xd),Y)
           => aa(A,$o,ord_less(A,Y),Xd) ) ) ) ).

% linorder_neqE
tff(fact_7392_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ~ aa(A,$o,ord_less(A,Y),Xd) ) ) ).

% order_less_asym
tff(fact_7393_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( ( Xd != Y )
        <=> ( aa(A,$o,ord_less(A,Xd),Y)
            | aa(A,$o,ord_less(A,Y),Xd) ) ) ) ).

% linorder_neq_iff
tff(fact_7394_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_7395_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( aa(A,$o,ord_less(A,Y),Z)
           => aa(A,$o,ord_less(A,Xd),Z) ) ) ) ).

% order_less_trans
tff(fact_7396_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( aa(B,$o,ord_less(B,B2),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,ord_less(B,X),Y2)
                 => aa(A,$o,ord_less(A,aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_7397_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,ord_less(A,X),Y2)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_7398_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A] : ~ aa(A,$o,ord_less(A,Xd),Xd) ) ).

% order_less_irrefl
tff(fact_7399_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,ord_less(A,A2),aa(B,A,F2,B2))
         => ( aa(B,$o,ord_less(B,B2),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,ord_less(B,X),Y2)
                 => aa(A,$o,ord_less(A,aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_7400_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(B,$o,ord_less(B,aa(A,B,F2,B2)),C2)
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,ord_less(A,X),Y2)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_7401_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ~ aa(A,$o,ord_less(A,Y),Xd) ) ) ).

% order_less_not_sym
tff(fact_7402_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A,P: $o] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( aa(A,$o,ord_less(A,Y),Xd)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_7403_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
          | ( Xd = Y )
          | aa(A,$o,ord_less(A,Y),Xd) ) ) ).

% linorder_less_linear
tff(fact_7404_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( Xd != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_7405_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( Y != Xd ) ) ) ).

% order_less_imp_not_eq2
tff(fact_7406_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ~ aa(A,$o,ord_less(A,Y),Xd) ) ) ).

% order_less_imp_not_less
tff(fact_7407_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),Nb)
    <=> case_nat($o,$false,ord_less_eq(nat,M),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_7408_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_yy(nat,fun(nat,nat),Nb),M) ).

% max_Suc1
tff(fact_7409_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_yz(nat,fun(nat,nat),Nb),M) ).

% max_Suc2
tff(fact_7410_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),I: nat,V: A] : list_update(A,aa(list(A),list(A),cons(A,Xd),Xs),I,V) = case_nat(list(A),aa(list(A),list(A),cons(A,V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_za(A,fun(list(A),fun(A,fun(nat,list(A)))),Xd),Xs),V),I) ).

% list_update.simps(2)
tff(fact_7411_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_7412_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Nb),one_one(nat))
     => ( num_of_nat(Nb) = one2 ) ) ).

% num_of_nat_One
tff(fact_7413_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_bg(nat,nat),aa(nat,nat,minus_minus(nat,M),Nb)) ).

% diff_Suc
tff(fact_7414_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(W))),Nb)
        <=> case_nat($o,$false,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),Nb) ) ) ).

% bit_numeral_rec(1)
tff(fact_7415_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),Nb)
        <=> case_nat($o,$true,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),Nb) ) ) ).

% bit_numeral_rec(2)
tff(fact_7416_min__Suc1,axiom,
    ! [Nb: nat,M: nat] : ord_min(nat,aa(nat,nat,suc,Nb),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_zb(nat,fun(nat,nat),Nb),M) ).

% min_Suc1
tff(fact_7417_min__Suc2,axiom,
    ! [M: nat,Nb: nat] : ord_min(nat,M,aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_zc(nat,fun(nat,nat),Nb),M) ).

% min_Suc2
tff(fact_7418_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_7419_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_7420_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_7421_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,ord_less(fun(A,B),F2),G)
        <=> ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),F2),G)
            & ~ aa(fun(A,B),$o,ord_less_eq(fun(A,B),G),F2) ) ) ) ).

% less_fun_def
tff(fact_7422_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),Y)
         => ( aa(A,$o,ord_less(A,Xd),Y)
            | ( Xd = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_7423_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),Y)
          | aa(A,$o,ord_less(A,Y),Xd) ) ) ).

% linorder_le_less_linear
tff(fact_7424_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(B,$o,ord_less_eq(B,aa(A,B,F2,B2)),C2)
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,ord_less(A,X),Y2)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_7425_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,ord_less(A,A2),aa(B,A,F2,B2))
         => ( aa(B,$o,ord_less_eq(B,B2),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,ord_less_eq(B,X),Y2)
                 => aa(A,$o,ord_less_eq(A,aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_7426_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( aa(B,$o,ord_less(B,aa(A,B,F2,B2)),C2)
           => ( ! [X: A,Y2: A] :
                  ( aa(A,$o,ord_less_eq(A,X),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,F2,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_7427_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,ord_less_eq(A,A2),aa(B,A,F2,B2))
         => ( aa(B,$o,ord_less(B,B2),C2)
           => ( ! [X: B,Y2: B] :
                  ( aa(B,$o,ord_less(B,X),Y2)
                 => aa(A,$o,ord_less(A,aa(B,A,F2,X)),aa(B,A,F2,Y2)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_7428_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),Z)
           => aa(A,$o,ord_less(A,Xd),Z) ) ) ) ).

% order_less_le_trans
tff(fact_7429_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),Y)
         => ( aa(A,$o,ord_less(A,Y),Z)
           => aa(A,$o,ord_less(A,Xd),Z) ) ) ) ).

% order_le_less_trans
tff(fact_7430_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( aa(A,$o,ord_less_eq(A,A2),B2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_7431_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
         => ( ( A2 != B2 )
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_7432_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => aa(A,$o,ord_less_eq(A,Xd),Y) ) ) ).

% order_less_imp_le
tff(fact_7433_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xd),Y)
        <=> aa(A,$o,ord_less_eq(A,Y),Xd) ) ) ).

% linorder_not_less
tff(fact_7434_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( ~ aa(A,$o,ord_less_eq(A,Xd),Y)
        <=> aa(A,$o,ord_less(A,Y),Xd) ) ) ).

% linorder_not_le
tff(fact_7435_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
        <=> ( aa(A,$o,ord_less_eq(A,Xd),Y)
            & ( Xd != Y ) ) ) ) ).

% order_less_le
tff(fact_7436_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),Y)
        <=> ( aa(A,$o,ord_less(A,Xd),Y)
            | ( Xd = Y ) ) ) ) ).

% order_le_less
tff(fact_7437_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,ord_less_eq(A,B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_7438_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,ord_less_eq(A,A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_7439_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
        <=> ( aa(A,$o,ord_less_eq(A,B2),A2)
            & ~ aa(A,$o,ord_less_eq(A,A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_7440_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,$o,ord_less_eq(A,C2),B2)
           => aa(A,$o,ord_less(A,C2),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_7441_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$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_7442_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,ord_less_eq(A,B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_7443_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less_eq(A,B2),A2)
        <=> ( aa(A,$o,ord_less(A,B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_7444_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xd: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
         => ( ! [W2: A] :
                ( aa(A,$o,ord_less(A,Xd),W2)
               => ( aa(A,$o,ord_less(A,W2),Y)
                 => aa(A,$o,ord_less_eq(A,W2),Z) ) )
           => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_7445_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),Xd)
         => ( ! [W2: A] :
                ( aa(A,$o,ord_less(A,Z),W2)
               => ( aa(A,$o,ord_less(A,W2),Xd)
                 => aa(A,$o,ord_less_eq(A,Y),W2) ) )
           => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_7446_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> ( aa(A,$o,ord_less_eq(A,A2),B2)
            & ~ aa(A,$o,ord_less_eq(A,B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_7447_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less_eq(A,B2),C2)
           => aa(A,$o,ord_less(A,A2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_7448_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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_7449_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> ( aa(A,$o,ord_less_eq(A,A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_7450_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less_eq(A,A2),B2)
        <=> ( aa(A,$o,ord_less(A,A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_7451_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xd: A] :
          ( ~ aa(A,$o,ord_less_eq(A,Y),Xd)
         => aa(A,$o,ord_less(A,Xd),Y) ) ) ).

% not_le_imp_less
tff(fact_7452_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xd),Y)
        <=> ( aa(A,$o,ord_less_eq(A,Xd),Y)
            & ~ aa(A,$o,ord_less_eq(A,Y),Xd) ) ) ) ).

% less_le_not_le
tff(fact_7453_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X: A] :
              ( aa(A,$o,ord_less(A,X),Y)
             => aa(A,$o,ord_less_eq(A,X),Z) )
         => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ).

% dense_le
tff(fact_7454_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X: A] :
              ( aa(A,$o,ord_less(A,Z),X)
             => aa(A,$o,ord_less_eq(A,Y),X) )
         => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ).

% dense_ge
tff(fact_7455_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xd),Y)
         => ( ~ aa(A,$o,ord_less(A,Xd),Y)
          <=> ( Xd = Y ) ) ) ) ).

% antisym_conv2
tff(fact_7456_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xd: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xd),Y)
         => ( aa(A,$o,ord_less_eq(A,Xd),Y)
          <=> ( Xd = Y ) ) ) ) ).

% antisym_conv1
tff(fact_7457_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,ord_less(A,A2),B2)
        <=> ( ~ aa(A,$o,ord_less_eq(A,A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_7458_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xd: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xd),Y)
         => aa(A,$o,ord_less_eq(A,Y),Xd) ) ) ).

% leI
tff(fact_7459_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xd: A] :
          ( aa(A,$o,ord_less_eq(A,Y),Xd)
         => ~ aa(A,$o,ord_less(A,Xd),Y) ) ) ).

% leD
tff(fact_7460_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_7461_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_7462_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_7463_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_7464_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_7465_set__bits__int__unfold_H,axiom,
    ! [F2: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,F2) = $ite(
        ? [N6: nat] :
        ! [N10: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N6),N10)
         => ~ aa(nat,$o,F2,N10) ),
        groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list(nat),list($o),map(nat,$o,F2),upt(zero_zero(nat),ord_Least(nat,aTP_Lamp_zd(fun(nat,$o),fun(nat,$o),F2))))),
        $ite(
          ? [N6: nat] :
          ! [N10: nat] :
            ( aa(nat,$o,ord_less_eq(nat,N6),N10)
           => aa(nat,$o,F2,N10) ),
          $let(
            n2: nat,
            n2:= ord_Least(nat,aTP_Lamp_ze(fun(nat,$o),fun(nat,$o),F2)),
            aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list($o),list($o),aa(list($o),fun(list($o),list($o)),append($o),aa(list(nat),list($o),map(nat,$o,F2),upt(zero_zero(nat),n2))),aa(list($o),list($o),cons($o,$true),nil($o))))) ),
          zero_zero(int) ) ) ).

% set_bits_int_unfold'
tff(fact_7466_Eps__Opt__def,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      eps_Opt(A,P) = $ite(
        ? [X8: A] : aa(A,$o,P,X8),
        aa(A,option(A),some(A),fChoice(A,P)),
        none(A) ) ).

% Eps_Opt_def
tff(fact_7467_append_Oassoc,axiom,
    ! [A: $tType,A2: list(A),B2: list(A),C2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),B2)),C2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),B2),C2)) ).

% append.assoc
tff(fact_7468_append__assoc,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs3: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3)) ).

% append_assoc
tff(fact_7469_append__same__eq,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A),Zs3: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs3),Xs) )
    <=> ( Ys = Zs3 ) ) ).

% append_same_eq
tff(fact_7470_same__append__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs3: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs3) )
    <=> ( Ys = Zs3 ) ) ).

% same_append_eq
tff(fact_7471_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_7472_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_7473_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xd: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Xd),nil(A))),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xd ).

% concat_inth
tff(fact_7474_pure__false,axiom,
    pure_assn($false) = bot_bot(assn) ).

% pure_false
tff(fact_7475_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = bot_bot(assn) )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_7476_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_7477_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_7478_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_7479_list_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),A2) = nil(A) )
    <=> ( A2 = nil(B) ) ) ).

% list.map_disc_iff
tff(fact_7480_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),map(B,A,F2),Xs) )
    <=> ( Xs = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_7481_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = nil(A) )
    <=> ( Xs = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_7482_append_Oright__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),nil(A)) = A2 ).

% append.right_neutral
tff(fact_7483_append__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),nil(A)) = Xs ).

% append_Nil2
tff(fact_7484_append__self__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Xs )
    <=> ( Ys = nil(A) ) ) ).

% append_self_conv
tff(fact_7485_self__append__conv,axiom,
    ! [A: $tType,Y: list(A),Ys: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Y),Ys) )
    <=> ( Ys = nil(A) ) ) ).

% self_append_conv
tff(fact_7486_append__self__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Ys )
    <=> ( Xs = nil(A) ) ) ).

% append_self_conv2
tff(fact_7487_self__append__conv2,axiom,
    ! [A: $tType,Y: list(A),Xs: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Y) )
    <=> ( Xs = nil(A) ) ) ).

% self_append_conv2
tff(fact_7488_Nil__is__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_is_append_conv
tff(fact_7489_append__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = nil(A) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% append_is_Nil_conv
tff(fact_7490_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),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_7491_map__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F2),Xs)),aa(list(B),list(A),map(B,A,F2),Ys)) ).

% map_append
tff(fact_7492_list__update__nonempty,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xd: A] :
      ( ( list_update(A,Xs,K,Xd) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% list_update_nonempty
tff(fact_7493_concat__replicate__trivial,axiom,
    ! [A: $tType,I: nat] : concat(A,replicate(list(A),I,nil(A))) = nil(A) ).

% concat_replicate_trivial
tff(fact_7494_foldr__append,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Xs: list(B),Ys: list(B),A2: A] : aa(A,A,foldr(B,A,F2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),A2) = aa(A,A,foldr(B,A,F2,Xs),aa(A,A,foldr(B,A,F2,Ys),A2)) ).

% foldr_append
tff(fact_7495_concat__append,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xs)),concat(A,Ys)) ).

% concat_append
tff(fact_7496_ent__false__iff,axiom,
    ! [P: assn] :
      ( entails(P,bot_bot(assn))
    <=> ! [H2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P),H2) ) ).

% ent_false_iff
tff(fact_7497_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),Xd: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,P3),Xd)),aa(list(A),assn,snga_assn(A,P3),Y)) = bot_bot(assn) ) ).

% snga_same_false
tff(fact_7498_listI__assn__finite,axiom,
    ! [B: $tType,A: $tType,I6: set(nat),A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),I6)
     => ( vEBT_List_listI_assn(A,B,I6,A3,Xs,Xsi) = bot_bot(assn) ) ) ).

% listI_assn_finite
tff(fact_7499_some__opt__false__trivial,axiom,
    ! [A: $tType] : eps_Opt(A,aTP_Lamp_yl(A,$o)) = none(A) ).

% some_opt_false_trivial
tff(fact_7500_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_7501_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_7502_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_7503_append1__eq__conv,axiom,
    ! [A: $tType,Xs: list(A),Xd: A,Ys: list(A),Y: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xd),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),nil(A))) )
    <=> ( ( Xs = Ys )
        & ( Xd = Y ) ) ) ).

% append1_eq_conv
tff(fact_7504_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(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_7505_replicate__empty,axiom,
    ! [A: $tType,Nb: nat,Xd: A] :
      ( ( replicate(A,Nb,Xd) = nil(A) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_7506_empty__replicate,axiom,
    ! [A: $tType,Nb: nat,Xd: A] :
      ( ( nil(A) = replicate(A,Nb,Xd) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_7507_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A] : groups4207007520872428315er_sum(B,A,F2,A2,nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_7508_size__list__append,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F2),Xs)),aa(list(A),nat,size_list(A,F2),Ys)) ).

% size_list_append
tff(fact_7509_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_7510_list__assn__aux__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),L2: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,nil(A)),L2) = pure_assn(L2 = nil(B)) ).

% list_assn_aux_simps(1)
tff(fact_7511_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( concat(A,Xss) = nil(A) )
    <=> ! [X5: list(A)] :
          ( member(list(A),X5,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X5 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_7512_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = concat(A,Xss) )
    <=> ! [X5: list(A)] :
          ( member(list(A),X5,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X5 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_7513_Eps__Opt__eq__None,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( eps_Opt(A,P) = none(A) )
    <=> ~ ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% Eps_Opt_eq_None
tff(fact_7514_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_7515_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xd: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xd),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xd ).

% nth_append_length
tff(fact_7516_nth__append__first,axiom,
    ! [A: $tType,I: nat,L: list(A),L2: list(A)] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L2)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_append_first
tff(fact_7517_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),aa(list(A),fun(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_7518_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xd: A,Ys: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xd),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Y),Ys)) ).

% list_update_length
tff(fact_7519_list__assn__aux__append,axiom,
    ! [A: $tType,B: $tType,L1: list(A),L12: list(B),P: fun(A,fun(B,assn)),L22: list(A),L23: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L12) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L22)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L12),L23)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L1),L12)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L22),L23)) ) ) ).

% list_assn_aux_append
tff(fact_7520_list__assn__aux__append2,axiom,
    ! [A: $tType,B: $tType,L22: list(A),L23: list(B),P: fun(A,fun(B,assn)),L1: list(A),L12: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L22) = aa(list(B),nat,size_size(list(B)),L23) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L22)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L12),L23)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L1),L12)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L22),L23)) ) ) ).

% list_assn_aux_append2
tff(fact_7521_n__lists__Nil,axiom,
    ! [A: $tType,Nb: nat] :
      n_lists(A,Nb,nil(A)) = $ite(Nb = zero_zero(nat),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))),nil(list(A))) ).

% n_lists_Nil
tff(fact_7522_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_7523_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_zf(fun(B,A),fun(B,list(A)),F2)),Xs)) = aa(list(B),list(A),map(B,A,F2),Xs) ).

% concat_map_singleton
tff(fact_7524_hash__code__list__simps_I1_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32)] : hash_hash_code_list(A,H_a,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_7525_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_7526_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_7527_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_7528_list__assn_Oelims,axiom,
    ! [B: $tType,A: $tType,Xd: fun(A,fun(B,assn)),Xaa: list(A),Xba: list(B),Y: assn] :
      ( ( aa(list(B),assn,vEBT_List_list_assn(A,B,Xd,Xaa),Xba) = Y )
     => ( ( ( Xaa = nil(A) )
         => ( ( Xba = nil(B) )
           => ( Y != one_one(assn) ) ) )
       => ( ! [A5: A,As: list(A)] :
              ( ( Xaa = aa(list(A),list(A),cons(A,A5),As) )
             => ! [C4: B,Cs2: list(B)] :
                  ( ( Xba = aa(list(B),list(B),cons(B,C4),Cs2) )
                 => ( Y != aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Xd,A5),C4)),aa(list(B),assn,vEBT_List_list_assn(A,B,Xd,As),Cs2)) ) ) )
         => ( ( ? [V4: A,Va: list(A)] : Xaa = aa(list(A),list(A),cons(A,V4),Va)
             => ( ( Xba = nil(B) )
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ( Xaa = nil(A) )
               => ( ? [V4: B,Va: list(B)] : Xba = aa(list(B),list(B),cons(B,V4),Va)
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_7529_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_7530_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_7531_VEBT__internal_Oreplicatei_Osimps_I1_J,axiom,
    ! [A: $tType,Xd: heap_Time_Heap(A)] : vEBT_VEBT_replicatei(A,zero_zero(nat),Xd) = heap_Time_return(list(A),nil(A)) ).

% VEBT_internal.replicatei.simps(1)
tff(fact_7532_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xd: fun(A,nat)] : aa(list(A),nat,size_list(A,Xd),nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_7533_replicate__0,axiom,
    ! [A: $tType,Xd: A] : replicate(A,zero_zero(nat),Xd) = nil(A) ).

% replicate_0
tff(fact_7534_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_7535_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_7536_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_7537_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : subseqs(A,nil(A)) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% subseqs.simps(1)
tff(fact_7538_concat__eq__append__conv,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs3: list(A)] :
      ( ( concat(A,Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3) )
    <=> $ite(
          Xss = nil(list(A)),
          ( ( Ys = nil(A) )
          & ( Zs3 = nil(A) ) ),
          ? [Xss1: list(list(A)),Xs4: list(A),Xs5: list(A),Xss2: list(list(A))] :
            ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss1),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),Xs5)),Xss2)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss1)),Xs4) )
            & ( Zs3 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs5),concat(A,Xss2)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_7539_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,Xs: list(A),Ys: list(B),Zs3: list(C),Ws: list(D6),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D6),$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)),Zs3) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs3) = aa(list(D6),nat,size_size(list(D6)),Ws) )
         => ( aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,nil(A)),nil(B)),nil(C)),nil(D6))
           => ( ! [X: A,Xs2: list(A),Y2: B,Ys4: list(B),Z2: C,Zs2: list(C),W2: D6,Ws2: list(D6)] :
                  ( ( 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(D6),nat,size_size(list(D6)),Ws2) )
                     => ( aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,Xs2),Ys4),Zs2),Ws2)
                       => aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,aa(list(A),list(A),cons(A,X),Xs2)),aa(list(B),list(B),cons(B,Y2),Ys4)),aa(list(C),list(C),cons(C,Z2),Zs2)),aa(list(D6),list(D6),cons(D6,W2),Ws2)) ) ) ) )
             => aa(list(D6),$o,aa(list(C),fun(list(D6),$o),aa(list(B),fun(list(C),fun(list(D6),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D6),$o))),P,Xs),Ys),Zs3),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_7540_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs3: list(C),P: fun(list(A),fun(list(B),fun(list(C),$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)),Zs3) )
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X: A,Xs2: list(A),Y2: B,Ys4: list(B),Z2: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys4) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys4) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs2),Ys4),Zs2)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,aa(list(A),list(A),cons(A,X),Xs2)),aa(list(B),list(B),cons(B,Y2),Ys4)),aa(list(C),list(C),cons(C,Z2),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs),Ys),Zs3) ) ) ) ) ).

% list_induct3
tff(fact_7541_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))
       => ( ! [X: A,Xs2: list(A),Y2: B,Ys4: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys4) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ys4)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),cons(A,X),Xs2)),aa(list(B),list(B),cons(B,Y2),Ys4)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_7542_replicate__app__Cons__same,axiom,
    ! [A: $tType,Nb: nat,Xd: A,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xd)),aa(list(A),list(A),cons(A,Xd),Xs)) = aa(list(A),list(A),cons(A,Xd),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xd)),Xs)) ).

% replicate_app_Cons_same
tff(fact_7543_list__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X: A] : aa(list(A),$o,P,aa(list(A),list(A),cons(A,X),nil(A)))
       => ( ! [X: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P,Xs2)
               => aa(list(A),$o,P,aa(list(A),list(A),cons(A,X),Xs2)) ) )
         => aa(list(A),$o,P,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_7544_rev__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X: A] : aa(list(A),$o,P,aa(list(A),list(A),cons(A,X),nil(A)))
       => ( ! [X: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P,Xs2)
               => aa(list(A),$o,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),cons(A,X),nil(A)))) ) )
         => aa(list(A),$o,P,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_7545_append__eq__Cons__conv,axiom,
    ! [A: $tType,Ys: list(A),Zs3: list(A),Xd: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3) = aa(list(A),list(A),cons(A,Xd),Xs) )
    <=> ( ( ( Ys = nil(A) )
          & ( Zs3 = aa(list(A),list(A),cons(A,Xd),Xs) ) )
        | ? [Ys5: list(A)] :
            ( ( Ys = aa(list(A),list(A),cons(A,Xd),Ys5) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Zs3) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_7546_Cons__eq__append__conv,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Ys: list(A),Zs3: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xd),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3) )
    <=> ( ( ( Ys = nil(A) )
          & ( aa(list(A),list(A),cons(A,Xd),Xs) = Zs3 ) )
        | ? [Ys5: list(A)] :
            ( ( aa(list(A),list(A),cons(A,Xd),Ys5) = Ys )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Zs3) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_7547_Cons__eq__appendI,axiom,
    ! [A: $tType,Xd: A,Xs1: list(A),Ys: list(A),Xs: list(A),Zs3: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xd),Xs1) = Ys )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Zs3) )
       => ( aa(list(A),list(A),cons(A,Xd),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3) ) ) ) ).

% Cons_eq_appendI
tff(fact_7548_list__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))
     => ( ! [X: A,Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),cons(A,X),Xs2)),nil(B))
       => ( ! [Y2: B,Ys4: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),aa(list(B),list(B),cons(B,Y2),Ys4))
         => ( ! [X: A,Xs2: list(A),Y2: 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),cons(A,X),Xs2)),aa(list(B),list(B),cons(B,Y2),Ys4)) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_7549_neq__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
    <=> ? [Y5: A,Ys3: list(A)] : Xs = aa(list(A),list(A),cons(A,Y5),Ys3) ) ).

% neq_Nil_conv
tff(fact_7550_rev__exhaust,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ~ ! [Ys4: list(A),Y2: A] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Y2),nil(A))) ) ).

% rev_exhaust
tff(fact_7551_rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [X: A,Xs2: list(A)] :
            ( aa(list(A),$o,P,Xs2)
           => aa(list(A),$o,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),cons(A,X),nil(A)))) )
       => aa(list(A),$o,P,Xs) ) ) ).

% rev_induct
tff(fact_7552_append__Cons,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Xd),Xs)),Ys) = aa(list(A),list(A),cons(A,Xd),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% append_Cons
tff(fact_7553_remdups__adj_Ocases,axiom,
    ! [A: $tType,Xd: list(A)] :
      ( ( Xd != nil(A) )
     => ( ! [X: A] : Xd != aa(list(A),list(A),cons(A,X),nil(A))
       => ~ ! [X: A,Y2: A,Xs2: list(A)] : Xd != aa(list(A),list(A),cons(A,X),aa(list(A),list(A),cons(A,Y2),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_7554_transpose_Ocases,axiom,
    ! [A: $tType,Xd: list(list(A))] :
      ( ( Xd != nil(list(A)) )
     => ( ! [Xss3: list(list(A))] : Xd != aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss3)
       => ~ ! [X: A,Xs2: list(A),Xss3: list(list(A))] : Xd != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss3) ) ) ).

% transpose.cases
tff(fact_7555_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xd: list(A)] :
          ( ! [X: A,Xs2: list(A)] : Xd != aa(list(A),list(A),cons(A,X),Xs2)
         => ( Xd = nil(A) ) ) ) ).

% min_list.cases
tff(fact_7556_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Y != aa(list(A),list(A),cons(A,X212),X223) ) ).

% list.exhaust
tff(fact_7557_list_OdiscI,axiom,
    ! [A: $tType,List: list(A),X21: A,X222: list(A)] :
      ( ( List = aa(list(A),list(A),cons(A,X21),X222) )
     => ( List != nil(A) ) ) ).

% list.discI
tff(fact_7558_list_Odistinct_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : nil(A) != aa(list(A),list(A),cons(A,X21),X222) ).

% list.distinct(1)
tff(fact_7559_list__assn_Ocases,axiom,
    ! [A: $tType,B: $tType,Xd: product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))] :
      ( ! [P4: fun(A,fun(B,assn))] : Xd != aa(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)),product_Pair(list(A),list(B),nil(A)),nil(B)))
     => ( ! [P4: fun(A,fun(B,assn)),A5: A,As: list(A),C4: B,Cs2: list(B)] : Xd != aa(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)),product_Pair(list(A),list(B),aa(list(A),list(A),cons(A,A5),As)),aa(list(B),list(B),cons(B,C4),Cs2)))
       => ( ! [Uu2: fun(A,fun(B,assn)),V4: A,Va: list(A)] : Xd != aa(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)),product_Pair(list(A),list(B),aa(list(A),list(A),cons(A,V4),Va)),nil(B)))
         => ~ ! [Uu2: fun(A,fun(B,assn)),V4: B,Va: list(B)] : Xd != aa(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)),product_Pair(list(A),list(B),nil(A)),aa(list(B),list(B),cons(B,V4),Va))) ) ) ) ).

% list_assn.cases
tff(fact_7560_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,Xd: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F4: fun(A,B),Bs2: list(B)] : Xd != aa(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)),F4),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),nil(A)),Bs2))
     => ~ ! [F4: fun(A,B),A5: A,As: list(A),Bs2: list(B)] : Xd != aa(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)),F4),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),aa(list(A),list(A),cons(A,A5),As)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_7561_successively_Ocases,axiom,
    ! [A: $tType,Xd: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P4: fun(A,fun(A,$o))] : Xd != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),product_Pair(fun(A,fun(A,$o)),list(A),P4),nil(A))
     => ( ! [P4: fun(A,fun(A,$o)),X: A] : Xd != aa(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),cons(A,X),nil(A)))
       => ~ ! [P4: fun(A,fun(A,$o)),X: A,Y2: A,Xs2: list(A)] : Xd != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),product_Pair(fun(A,fun(A,$o)),list(A),P4),aa(list(A),list(A),cons(A,X),aa(list(A),list(A),cons(A,Y2),Xs2))) ) ) ).

% successively.cases
tff(fact_7562_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xd: product_prod(fun(A,B),list(A))] :
          ( ! [F4: fun(A,B),X: A] : Xd != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),F4),aa(list(A),list(A),cons(A,X),nil(A)))
         => ( ! [F4: fun(A,B),X: A,Y2: A,Zs2: list(A)] : Xd != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),F4),aa(list(A),list(A),cons(A,X),aa(list(A),list(A),cons(A,Y2),Zs2)))
           => ~ ! [A5: fun(A,B)] : Xd != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),A5),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_7563_sorted__wrt_Ocases,axiom,
    ! [A: $tType,Xd: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P4: fun(A,fun(A,$o))] : Xd != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),product_Pair(fun(A,fun(A,$o)),list(A),P4),nil(A))
     => ~ ! [P4: fun(A,fun(A,$o)),X: A,Ys4: list(A)] : Xd != aa(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),cons(A,X),Ys4)) ) ).

% sorted_wrt.cases
tff(fact_7564_shuffles_Ocases,axiom,
    ! [A: $tType,Xd: product_prod(list(A),list(A))] :
      ( ! [Ys4: list(A)] : Xd != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys4)
     => ( ! [Xs2: list(A)] : Xd != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs2),nil(A))
       => ~ ! [X: A,Xs2: list(A),Y2: A,Ys4: list(A)] : Xd != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),cons(A,X),Xs2)),aa(list(A),list(A),cons(A,Y2),Ys4)) ) ) ).

% shuffles.cases
tff(fact_7565_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,Nb: nat,Xd: A] : replicate(A,aa(nat,nat,suc,Nb),Xd) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xd)),aa(list(A),list(A),cons(A,Xd),nil(A))) ).

% replicate_Suc_conv_snoc
tff(fact_7566_replicate__append__same,axiom,
    ! [A: $tType,I: nat,Xd: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,I,Xd)),aa(list(A),list(A),cons(A,Xd),nil(A))) = aa(list(A),list(A),cons(A,Xd),replicate(A,I,Xd)) ).

% replicate_append_same
tff(fact_7567_product__lists_Osimps_I1_J,axiom,
    ! [A: $tType] : product_lists(A,nil(list(A))) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% product_lists.simps(1)
tff(fact_7568_List_Omap__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(A),Zs3: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3) )
    <=> ? [Us2: list(B),Vs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
          & ( Ys = aa(list(B),list(A),map(B,A,F2),Us2) )
          & ( Zs3 = aa(list(B),list(A),map(B,A,F2),Vs2) ) ) ) ).

% List.map_eq_append_conv
tff(fact_7569_List_Oappend__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list(A),Zs3: list(A),F2: fun(B,A),Xs: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3) = aa(list(B),list(A),map(B,A,F2),Xs) )
    <=> ? [Us2: list(B),Vs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
          & ( Ys = aa(list(B),list(A),map(B,A,F2),Us2) )
          & ( Zs3 = aa(list(B),list(A),map(B,A,F2),Vs2) ) ) ) ).

% List.append_eq_map_conv
tff(fact_7570_list_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(list(B),list(A),map(B,A,F2),nil(B)) = nil(A) ).

% list.simps(8)
tff(fact_7571_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xd: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xd),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_7572_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),Y5: A] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% length_Suc_rev_conv
tff(fact_7573_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),X: A,Xs6: list(A),Y2: A,Ys6: list(A)] :
            ( ( X != Y2 )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,X),nil(A))),Xs6)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Y2),nil(A))),Ys6)) ) ) ) ) ).

% same_length_different
tff(fact_7574_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) )
    <=> ? [Y5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% length_Suc_conv_rev
tff(fact_7575_concat_Osimps_I2_J,axiom,
    ! [A: $tType,Xd: list(A),Xs: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),cons(list(A),Xd),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xd),concat(A,Xs)) ).

% concat.simps(2)
tff(fact_7576_append__replicate__commute,axiom,
    ! [A: $tType,Nb: nat,Xd: A,K: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xd)),replicate(A,K,Xd)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,K,Xd)),replicate(A,Nb,Xd)) ).

% append_replicate_commute
tff(fact_7577_product_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Uu: list(B)] : product(A,B,nil(A),Uu) = nil(product_prod(A,B)) ).

% product.simps(1)
tff(fact_7578_append__Nil,axiom,
    ! [A: $tType,Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) = Ys ).

% append_Nil
tff(fact_7579_append_Oleft__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),A2) = A2 ).

% append.left_neutral
tff(fact_7580_eq__Nil__appendI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_7581_append__eq__appendI,axiom,
    ! [A: $tType,Xs: list(A),Xs1: list(A),Zs3: list(A),Ys: list(A),Us: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Xs1) = Zs3 )
     => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Us) )
       => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs3),Us) ) ) ) ).

% append_eq_appendI
tff(fact_7582_append__eq__append__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs3: list(A),Ts: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs3),Ts) )
    <=> ? [Us2: list(A)] :
          ( ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs3),Us2) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ys) = Ts ) )
          | ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2) = Zs3 )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_7583_list__update__code_I1_J,axiom,
    ! [A: $tType,I: nat,Y: A] : list_update(A,nil(A),I,Y) = nil(A) ).

% list_update_code(1)
tff(fact_7584_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,I: nat,V: A] : list_update(A,nil(A),I,V) = nil(A) ).

% list_update.simps(1)
tff(fact_7585_comm__append__are__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
     => ? [M2: nat,N: nat,Zs2: list(A)] :
          ( ( concat(A,replicate(list(A),M2,Zs2)) = Xs )
          & ( concat(A,replicate(list(A),N,Zs2)) = Ys ) ) ) ).

% comm_append_are_replicate
tff(fact_7586_concat__conv__foldr,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,Xss) = aa(list(A),list(A),foldr(list(A),list(A),append(A),Xss),nil(A)) ).

% concat_conv_foldr
tff(fact_7587_concat_Osimps_I1_J,axiom,
    ! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).

% concat.simps(1)
tff(fact_7588_split__list,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys4: list(A),Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Xd),Zs2)) ) ).

% split_list
tff(fact_7589_split__list__last,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys4: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Xd),Zs2)) )
          & ~ member(A,Xd,aa(list(A),set(A),set2(A),Zs2)) ) ) ).

% split_list_last
tff(fact_7590_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ? [Ys4: list(A),X: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X),Zs2))
          & aa(A,$o,P,X) ) ) ).

% split_list_prop
tff(fact_7591_split__list__first,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys4: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Xd),Zs2)) )
          & ~ member(A,Xd,aa(list(A),set(A),set2(A),Ys4)) ) ) ).

% split_list_first
tff(fact_7592_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ~ ! [Ys4: list(A),X: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X),Zs2))
           => ~ aa(A,$o,P,X) ) ) ).

% split_list_propE
tff(fact_7593_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Ys: list(A),Xs3: list(A),Ys7: list(A)] :
      ( ~ member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ( ~ member(A,Xd,aa(list(A),set(A),set2(A),Ys))
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xd),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),cons(A,Xd),Ys7)) )
        <=> ( ( Xs = Xs3 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_7594_in__set__conv__decomp,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys3: list(A),Zs: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Xd),Zs)) ) ).

% in_set_conv_decomp
tff(fact_7595_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ? [Ys4: list(A),X: A,Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X),Zs2)) )
          & aa(A,$o,P,X)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_last_prop
tff(fact_7596_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ? [Ys4: list(A),X: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X),Zs2))
          & aa(A,$o,P,X)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_first_prop
tff(fact_7597_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ~ ! [Ys4: list(A),X: A,Zs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X),Zs2)) )
           => ( aa(A,$o,P,X)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_last_propE
tff(fact_7598_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ~ ! [Ys4: list(A),X: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X),Zs2))
           => ( aa(A,$o,P,X)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Ys4))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_first_propE
tff(fact_7599_in__set__conv__decomp__last,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys3: list(A),Zs: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Xd),Zs)) )
          & ~ member(A,Xd,aa(list(A),set(A),set2(A),Zs)) ) ) ).

% in_set_conv_decomp_last
tff(fact_7600_in__set__conv__decomp__first,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys3: list(A),Zs: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Xd),Zs)) )
          & ~ member(A,Xd,aa(list(A),set(A),set2(A),Ys3)) ) ) ).

% in_set_conv_decomp_first
tff(fact_7601_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X5) )
    <=> ? [Ys3: list(A),X5: A,Zs: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X5),Zs)) )
          & aa(A,$o,P,X5)
          & ! [Xa3: A] :
              ( member(A,Xa3,aa(list(A),set(A),set2(A),Zs))
             => ~ aa(A,$o,P,Xa3) ) ) ) ).

% split_list_last_prop_iff
tff(fact_7602_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X5) )
    <=> ? [Ys3: list(A),X5: A] :
          ( ? [Zs: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X5),Zs))
          & aa(A,$o,P,X5)
          & ! [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_7603_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_7604_ent__false,axiom,
    ! [P: assn] : entails(bot_bot(assn),P) ).

% ent_false
tff(fact_7605_mod__false,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(bot_bot(assn)),H) ).

% mod_false
tff(fact_7606_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_7607_list__assn__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A2: A,As2: list(A)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),cons(A,A2),As2)),nil(B)) = bot_bot(assn) ).

% list_assn_simps(3)
tff(fact_7608_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),cons(B,C2),Cs)) = bot_bot(assn) ).

% list_assn_simps(4)
tff(fact_7609_list__assn_Osimps_I3_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),V: A,Va2: list(A)] : aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,aa(list(A),list(A),cons(A,V),Va2)),nil(B)) = bot_bot(assn) ).

% list_assn.simps(3)
tff(fact_7610_list__assn_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),V: B,Va2: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,nil(A)),aa(list(B),list(B),cons(B,V),Va2)) = bot_bot(assn) ).

% list_assn.simps(4)
tff(fact_7611_replicate__add,axiom,
    ! [A: $tType,Nb: nat,M: nat,Xd: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M),Xd) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xd)),replicate(A,M,Xd)) ).

% replicate_add
tff(fact_7612_len__greater__imp__nonempty,axiom,
    ! [A: $tType,Xd: nat,L: list(A)] :
      ( aa(nat,$o,ord_less(nat,Xd),aa(list(A),nat,size_size(list(A)),L))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_7613_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N: nat,Zs2: list(A)] :
              ( aa(nat,$o,ord_less(nat,one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_7614_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys: list(A),Xd: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),I,Xd) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,I,Xd)),Ys) ) ) ).

% list_update_append1
tff(fact_7615_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A5: A] : L = aa(list(A),list(A),cons(A,A5),nil(A)) ) ).

% list_decomp_1
tff(fact_7616_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option(product_prod(nat,nat)),Va2: 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,Va2,Vb,Vc2)) = bot_bot(assn) ).

% vebt_assn_raw.simps(4)
tff(fact_7617_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(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_7618_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat,Xd: A] :
      list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),Nb,Xd) = $ite(aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,Nb,Xd)),Ys),aa(list(A),list(A),aa(list(A),fun(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)),Xd))) ).

% list_update_append
tff(fact_7619_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,I),K)
     => ( aa(nat,$o,ord_less_eq(nat,K),aa(list(A),nat,size_size(list(A)),Xs))
       => $let(
            p: nat,
            p:= aa(list(A),nat,size_size(list(A)),Ys),
            slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),p),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),p),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs)) ) ) ) ).

% slice_prepend
tff(fact_7620_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups4207007520872428315er_sum(B,A,F2,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,F2,A2,Ys))) ) ).

% horner_sum_append
tff(fact_7621_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)) )
     => ? [A5: A,B3: A] : L = aa(list(A),list(A),cons(A,A5),aa(list(A),list(A),cons(A,B3),nil(A))) ) ).

% list_decomp_2
tff(fact_7622_vebt__assn__raw_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xd),Xaa) = Y )
     => ( ! [A5: $o,B3: $o] :
            ( ( Xd = vEBT_Leaf((A5),(B3)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != pure_assn(( ( (Ai) = (A5) )
                      & ( (Bi) = (B3) ) )) ) ) )
       => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xd = vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2) )
             => ! [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,Summary2),Summaryi2))),
                        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_kv(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list2),Tree_array2))) ) ) )
         => ( ( ? [V4: option(product_prod(nat,nat)),Va: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT] : Xd = vEBT_Node(V4,Va,Vb3,Vc3)
             => ( ? [Vd3: $o,Ve3: $o] : Xaa = vEBT_Leafi((Vd3),(Ve3))
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : Xd = vEBT_Leaf((Vd3),(Ve3))
               => ( ? [V4: option(product_prod(nat,nat)),Va: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] : Xaa = vEBT_Nodei(V4,Va,Vb3,Vc3)
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_7623_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( aa(char,$o,ord_less_eq(char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))),char2((C0),(C1),(C22),(C32),(C42),(C52),(C62),(C7)))
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,foldr($o,nat,aTP_Lamp_zg($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B1)),aa(list($o),list($o),cons($o,(B22)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o)))))))))),zero_zero(nat))),aa(nat,nat,foldr($o,nat,aTP_Lamp_zg($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(C0)),aa(list($o),list($o),cons($o,(C1)),aa(list($o),list($o),cons($o,(C22)),aa(list($o),list($o),cons($o,(C32)),aa(list($o),list($o),cons($o,(C42)),aa(list($o),list($o),cons($o,(C52)),aa(list($o),list($o),cons($o,(C62)),aa(list($o),list($o),cons($o,(C7)),nil($o)))))))))),zero_zero(nat))) ) ).

% less_eq_char_simp
tff(fact_7624_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( vEBT_Leafi((X21),(X222)) = vEBT_Leafi((Y21),(Y222)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y222) ) ) ) ).

% VEBTi.inject(2)
tff(fact_7625_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_7626_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_7627_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,ord_less_eq(nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_7628_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),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_7629_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( aa(char,$o,ord_less(char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))),char2((C0),(C1),(C22),(C32),(C42),(C52),(C62),(C7)))
    <=> aa(nat,$o,ord_less(nat,aa(nat,nat,foldr($o,nat,aTP_Lamp_zg($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B1)),aa(list($o),list($o),cons($o,(B22)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o)))))))))),zero_zero(nat))),aa(nat,nat,foldr($o,nat,aTP_Lamp_zg($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(C0)),aa(list($o),list($o),cons($o,(C1)),aa(list($o),list($o),cons($o,(C22)),aa(list($o),list($o),cons($o,(C32)),aa(list($o),list($o),cons($o,(C42)),aa(list($o),list($o),cons($o,(C52)),aa(list($o),list($o),cons($o,(C62)),aa(list($o),list($o),cons($o,(C7)),nil($o)))))))))),zero_zero(nat))) ) ).

% less_char_simp
tff(fact_7630_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_7631_list__encode_Ocases,axiom,
    ! [Xd: list(nat)] :
      ( ( Xd != nil(nat) )
     => ~ ! [X: nat,Xs2: list(nat)] : Xd != aa(list(nat),list(nat),cons(nat,X),Xs2) ) ).

% list_encode.cases
tff(fact_7632_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option(product_prod(nat,nat)),Va2: 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,Va2,Vb,Vc2)),vEBT_Leafi((Vd),(Ve))) = bot_bot(assn) ).

% vebt_assn_raw.simps(3)
tff(fact_7633_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,ord_less_eq(nat,I),J),aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),cons(nat,J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_7634_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),cons(nat,J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_7635_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I)),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_7636_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(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_7637_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi,X21: $o,X222: $o] : vEBT_Nodei(X11,X12,X13,X14) != vEBT_Leafi((X21),(X222)) ).

% VEBTi.distinct(1)
tff(fact_7638_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: array(vEBT_VEBTi),X142: vEBT_VEBTi] : Y != vEBT_Nodei(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leafi((X212),(X223)) ) ).

% VEBTi.exhaust
tff(fact_7639_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V739175172307565963ildupi(zero_zero(nat)) = heap_Time_return(vEBT_VEBTi,vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(1)
tff(fact_7640_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Leafi((X21),(X222))) = zero_zero(nat) ).

% VEBTi.size(4)
tff(fact_7641_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Leafi((X21),(X222))) = zero_zero(nat) ).

% VEBTi.size_gen(2)
tff(fact_7642_concat__eq__appendD,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs3: list(A)] :
      ( ( concat(A,Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs3) )
     => ( ( Xss != nil(list(A)) )
       => ? [Xss12: list(list(A)),Xs2: list(A),Xs6: list(A),Xss22: list(list(A))] :
            ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss12),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),Xs6)),Xss22)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss12)),Xs2) )
            & ( Zs3 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs6),concat(A,Xss22)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_7643_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V739175172307565963ildupi(aa(nat,nat,suc,zero_zero(nat))) = heap_Time_return(vEBT_VEBTi,vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_7644_vebt__assn__raw_Ocases,axiom,
    ! [Xd: product_prod(vEBT_VEBT,vEBT_VEBTi)] :
      ( ! [A5: $o,B3: $o,Ai: $o,Bi: $o] : Xd != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Leaf((A5),(B3))),vEBT_Leafi((Ai),(Bi)))
     => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: vEBT_VEBTi] : Xd != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2)),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2))
       => ( ! [V4: option(product_prod(nat,nat)),Va: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : Xd != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Node(V4,Va,Vb3,Vc3)),vEBT_Leafi((Vd3),(Ve3)))
         => ~ ! [Vd3: $o,Ve3: $o,V4: option(product_prod(nat,nat)),Va: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] : Xd != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Leaf((Vd3),(Ve3))),vEBT_Nodei(V4,Va,Vb3,Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_7645_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      subseqs(A,aa(list(A),list(A),cons(A,Xd),Xs)) = $let(
        xss: list(list(A)),
        xss:= subseqs(A,Xs),
        aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),map(list(A),list(A),cons(A,Xd)),xss)),xss) ) ).

% subseqs.simps(2)
tff(fact_7646_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_7647_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xd: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),cons(A,Xd),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(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),product_Pair(A,B,Xd)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_7648_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,ord_less(nat,I),J),aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_7649_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),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),cons(nat,I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa(nat,nat,suc,I),H) )
        & aa(nat,$o,ord_less_eq(nat,L),I)
        & aa(nat,$o,ord_less(nat,I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_7650_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa($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($o,code_integer,zero_neq_one_of_bool(code_integer),(B22)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B1)))),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_7651_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : aa(char,A,comm_s6883823935334413003f_char(A),char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B1)),aa(list($o),list($o),cons($o,(B22)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_7652_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : 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_7653_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_7654_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_7655_range__nat__of__char,axiom,
    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_7656_char_Osize_I2_J,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : aa(char,nat,size_size(char),char2((X15),(X2),(X32),(X42),(X52),(X62),(X72),(X82))) = zero_zero(nat) ).

% char.size(2)
tff(fact_7657_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : aa(A,char,unique5772411509450598832har_of(A),Nb) = char2(~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),one_one(nat)),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(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_7658_char_Osize__gen,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : size_char(char2((X15),(X2),(X32),(X42),(X52),(X62),(X72),(X82))) = zero_zero(nat) ).

% char.size_gen
tff(fact_7659_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) )
        <=> ( modulo_modulo(A,M,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = 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_7660_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : aa(A,char,unique5772411509450598832har_of(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_7661_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)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ).

% of_char_of
tff(fact_7662_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_7663_UNIV__char__of__nat,axiom,
    top_top(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_7664_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,M: A] :
          ( aa(nat,$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_7665_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_7666_vebt__assn__raw_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xd),Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,Xd),Xaa))
       => ( ! [A5: $o,B3: $o] :
              ( ( Xd = vEBT_Leaf((A5),(B3)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = pure_assn(( ( (Ai) = (A5) )
                          & ( (Bi) = (B3) ) )) )
                   => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Leaf((A5),(B3))),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xd = vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2) )
               => ! [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,Summary2),Summaryi2))),
                            ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_kv(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list2),Tree_array2))) )
                     => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2)),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2))) ) ) )
           => ( ! [V4: option(product_prod(nat,nat)),Va: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT] :
                  ( ( Xd = vEBT_Node(V4,Va,Vb3,Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot(assn) )
                       => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Node(V4,Va,Vb3,Vc3)),vEBT_Leafi((Vd3),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xd = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V4: option(product_prod(nat,nat)),Va: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] :
                        ( ( Xaa = vEBT_Nodei(V4,Va,Vb3,Vc3) )
                       => ( ( Y = bot_bot(assn) )
                         => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),product_Pair(vEBT_VEBT,vEBT_VEBTi,vEBT_Leaf((Vd3),(Ve3))),vEBT_Nodei(V4,Va,Vb3,Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_7667_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_zi(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_7668_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),Xs)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),transpose(B,Xs)) ).

% transpose_map_map
tff(fact_7669_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_7670_transpose_Osimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss)) = transpose(A,Xss) ).

% transpose.simps(2)
tff(fact_7671_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X5: list(A)] :
          ( member(list(A),X5,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( X5 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_7672_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_zj(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_7673_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_7674_upto_Opelims,axiom,
    ! [Xd: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xd,Xaa) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xd),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,ord_less_eq(int,Xd),Xaa),aa(list(int),list(int),cons(int,Xd),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),one_one(int)),Xaa)),nil(int)) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xd),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_7675_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,ord_less(int,J),I) ) ).

% upto_Nil
tff(fact_7676_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,ord_less(int,J),I) ) ).

% upto_Nil2
tff(fact_7677_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,ord_less(int,J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_7678_upto__single,axiom,
    ! [I: int] : upto(I,I) = aa(list(int),list(int),cons(int,I),nil(int)) ).

% upto_single
tff(fact_7679_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_7680_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_7681_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,ord_less_eq(int,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb)),aa(list(int),list(int),cons(int,aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),Nb))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_7682_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,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),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_7683_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,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),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_7684_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,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),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_7685_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list(int)] : upto_aux(I,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),Js) ).

% upto_aux_def
tff(fact_7686_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil(int)) ).

% upto_code
tff(fact_7687_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_7688_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( aa(int,$o,ord_less_eq(int,J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(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_7689_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( aa(int,$o,ord_less_eq(int,J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(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_7690_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_7691_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_7692_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,ord_less_eq(int,I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_7693_upto_Oelims,axiom,
    ! [Xd: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xd,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,ord_less_eq(int,Xd),Xaa),aa(list(int),list(int),cons(int,Xd),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xd),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_7694_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_7695_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,minus_minus(int,J),one_one(int)))),aa(list(int),list(int),cons(int,J),nil(int))) ) ) ).

% upto_rec2
tff(fact_7696_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,minus_minus(int,J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_7697_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( aa(int,$o,ord_less_eq(int,J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,minus_minus(int,J),one_one(int)))),aa(list(int),list(int),cons(int,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_7698_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I),J))
     => ( upto(I,J) = $ite(aa(int,$o,ord_less_eq(int,I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_7699_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,insert(list(A),nil(A),bot_bot(set(list(A))))) = image(A,list(A),aTP_Lamp_zk(A,list(A)),A3) ).

% set_Cons_sing_Nil
tff(fact_7700_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),aa(list(nat),fun(list(nat),list(nat)),append(nat),linord4507533701916653071of_set(nat,set_ord_atMost(nat,K))),aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_7701_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A,bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_7702_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( linord4507533701916653071of_set(A,A3) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_7703_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,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_7704_sorted__list__of__set__range,axiom,
    ! [M: nat,Nb: nat] : linord4507533701916653071of_set(nat,set_or7035219750837199246ssThan(nat,M,Nb)) = upt(M,Nb) ).

% sorted_list_of_set_range
tff(fact_7705_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( linord4507533701916653071of_set(A,A3) = nil(A) )
          <=> ( A3 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7706_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),aa(list(nat),fun(list(nat),list(nat)),append(nat),linord4507533701916653071of_set(nat,set_ord_lessThan(nat,K))),aa(list(nat),list(nat),cons(nat,K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_7707_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( ( linord4507533701916653071of_set(A,A3) = linord4507533701916653071of_set(A,B4) )
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),$o,finite_finite2(A),B4)
             => ( A3 = B4 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_7708_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_7709_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_7710_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_zl(set(A),fun(set(list(A)),fun(list(A),$o)),A3),XS)) ).

% set_Cons_def
tff(fact_7711_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,minus_minus(nat,J),I))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_7712_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_7713_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = insert(list(A),nil(A),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_7714_list_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F12: A,F22: fun(B,fun(list(B),A)),X21: B,X222: list(B)] : aa(list(B),A,case_list(A,B,F12,F22),aa(list(B),list(B),cons(B,X21),X222)) = aa(list(B),A,aa(B,fun(list(B),A),F22,X21),X222) ).

% list.simps(5)
tff(fact_7715_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F12: B,F22: fun(C,fun(list(C),B)),List: list(C)] : aa(B,A,H,aa(list(C),B,case_list(B,C,F12,F22),List)) = aa(list(C),A,case_list(A,C,aa(B,A,H,F12),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_zm(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H),F22)),List) ).

% list.case_distrib
tff(fact_7716_list_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F12: A,F22: fun(B,fun(list(B),A))] : aa(list(B),A,case_list(A,B,F12,F22),nil(B)) = F12 ).

% list.simps(4)
tff(fact_7717_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A3: set(A),As3: list(set(A))] : listset(A,aa(list(set(A)),list(set(A)),cons(set(A),A3),As3)) = set_Cons(A,A3,listset(A,As3)) ).

% listset.simps(2)
tff(fact_7718_transpose_Oelims,axiom,
    ! [A: $tType,Xd: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xd) = Y )
     => ( ( ( Xd = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss3: list(list(A))] :
              ( ( Xd = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss3) )
             => ( Y != transpose(A,Xss3) ) )
         => ~ ! [X: A,Xs2: list(A),Xss3: list(list(A))] :
                ( ( Xd = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss3) )
               => ( Y != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_zn(A,fun(list(A),list(A))))),Xss3)))),transpose(A,aa(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_zo(A,fun(list(A),list(list(A)))))),Xss3))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_7719_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xd),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xd),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_zn(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(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_zo(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_7720_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xd: list(A),Y: A] :
          ( ( min_list(A,Xd) = Y )
         => ( ! [X: A,Xs2: list(A)] :
                ( ( Xd = aa(list(A),list(A),cons(A,X),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zp(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs2)),Xs2) ) )
           => ~ ( ( Xd = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_7721_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xd),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xd),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xd),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_zn(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(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_zo(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_7722_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_zq(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_7723_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_zr(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_7724_transpose_Opsimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss)) = transpose(A,Xss) ) ) ).

% transpose.psimps(2)
tff(fact_7725_transpose_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
     => ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).

% transpose.psimps(1)
tff(fact_7726_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),$o)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => aa(list(list(A)),$o,P,nil(list(A))) )
       => ( ! [Xss3: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss3))
             => ( aa(list(list(A)),$o,P,Xss3)
               => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss3)) ) )
         => ( ! [X: A,Xs2: list(A),Xss3: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss3))
               => ( aa(list(list(A)),$o,P,aa(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_zo(A,fun(list(A),list(list(A)))))),Xss3))))
                 => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss3)) ) )
           => aa(list(list(A)),$o,P,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_7727_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xd: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),cons(A,Xd),Xs)) = aa(list(A),A,case_list(A,A,Xd,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zp(A,fun(list(A),fun(A,fun(list(A),A))),Xd),Xs)),Xs) ) ).

% min_list.simps
tff(fact_7728_transpose_Opelims,axiom,
    ! [A: $tType,Xd: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xd) = Y )
     => ( accp(list(list(A)),transpose_rel(A),Xd)
       => ( ( ( Xd = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss3: list(list(A))] :
                ( ( Xd = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss3) )
               => ( ( Y = transpose(A,Xss3) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss3)) ) )
           => ~ ! [X: A,Xs2: list(A),Xss3: list(list(A))] :
                  ( ( Xd = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss3) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_zn(A,fun(list(A),list(A))))),Xss3)))),transpose(A,aa(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_zo(A,fun(list(A),list(list(A)))))),Xss3))))) )
                   => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss3)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_7729_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xd: list(A),Y: A] :
          ( ( min_list(A,Xd) = Y )
         => ( accp(list(A),min_list_rel(A),Xd)
           => ( ! [X: A,Xs2: list(A)] :
                  ( ( Xd = aa(list(A),list(A),cons(A,X),Xs2) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zp(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs2)),Xs2) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),cons(A,X),Xs2)) ) )
             => ~ ( ( Xd = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_7730_filter__upt__last,axiom,
    ! [A: $tType,P: fun(A,$o),L: list(A),Js: list(nat),J: nat,I: nat] :
      ( ( aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_zs(fun(A,$o),fun(list(A),fun(nat,$o)),P),L)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),cons(nat,J),nil(nat))) )
     => ( aa(nat,$o,ord_less(nat,J),I)
       => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
         => ~ aa(A,$o,P,aa(nat,A,nth(A,L),I)) ) ) ) ).

% filter_upt_last
tff(fact_7731_filter__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),filter2(A,Q),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_zt(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),Xs) ).

% filter_filter
tff(fact_7732_filter__True,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) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = Xs ) ) ).

% filter_True
tff(fact_7733_filter__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,P),Xs)),aa(list(A),list(A),filter2(A,P),Ys)) ).

% filter_append
tff(fact_7734_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs)) = collect(A,aa(list(A),fun(A,$o),aTP_Lamp_zu(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ).

% set_filter
tff(fact_7735_filter__False,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) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) ) ) ).

% filter_False
tff(fact_7736_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F2),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),filter2(B,comp(A,$o,B,P,F2)),Xs)) ).

% length_filter_map
tff(fact_7737_filter_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xd: A,Xs: list(A)] :
      aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),cons(A,Xd),Xs)) = $ite(aa(A,$o,P,Xd),aa(list(A),list(A),cons(A,Xd),aa(list(A),list(A),filter2(A,P),Xs)),aa(list(A),list(A),filter2(A,P),Xs)) ).

% filter.simps(2)
tff(fact_7738_Cons__eq__filterD,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xd),Xs) = aa(list(A),list(A),filter2(A,P),Ys) )
     => ? [Us3: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,Xd),Vs3)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X4) )
          & aa(A,$o,P,Xd)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% Cons_eq_filterD
tff(fact_7739_filter__eq__ConsD,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xd: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),cons(A,Xd),Xs) )
     => ? [Us3: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,Xd),Vs3)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X4) )
          & aa(A,$o,P,Xd)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% filter_eq_ConsD
tff(fact_7740_Cons__eq__filter__iff,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xd),Xs) = aa(list(A),list(A),filter2(A,P),Ys) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,Xd),Vs2)) )
          & ! [X5: A] :
              ( member(A,X5,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X5) )
          & aa(A,$o,P,Xd)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_7741_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xd: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),cons(A,Xd),Xs) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,Xd),Vs2)) )
          & ! [X5: A] :
              ( member(A,X5,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X5) )
          & aa(A,$o,P,Xd)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_7742_length__filter__less,axiom,
    ! [A: $tType,Xd: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xd)
       => aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_7743_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)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_kf(fun(A,$o),fun(A,$o),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_7744_replicate__length__filter,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,fequal(A,Xd)),Xs)),Xd) = aa(list(A),list(A),filter2(A,fequal(A,Xd)),Xs) ).

% replicate_length_filter
tff(fact_7745_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_7746_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P,X)
            <=> aa(A,$o,Q,X) ) )
       => ( aa(list(A),list(A),filter2(A,P),Xs) = aa(list(A),list(A),filter2(A,Q),Ys) ) ) ) ).

% filter_cong
tff(fact_7747_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = Xs )
    <=> ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X5) ) ) ).

% filter_id_conv
tff(fact_7748_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_7749_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F2,insert(A,Y,aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_zv(fun(A,B),fun(A,fun(A,$o)),F2),Y)),Xs) = aa(list(A),list(A),filter2(A,fequal(A,Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_7750_filter__concat,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(list(A))] : aa(list(A),list(A),filter2(A,P3),concat(A,Xs)) = concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),filter2(A,P3)),Xs)) ).

% filter_concat
tff(fact_7751_filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,comp(A,$o,B,P,F2)),Xs)) ).

% filter_map
tff(fact_7752_filter__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Nb: nat,Xd: A] :
      aa(list(A),list(A),filter2(A,P),replicate(A,Nb,Xd)) = $ite(aa(A,$o,P,Xd),replicate(A,Nb,Xd),nil(A)) ).

% filter_replicate
tff(fact_7753_filter_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(list(A),list(A),filter2(A,P),nil(A)) = nil(A) ).

% filter.simps(1)
tff(fact_7754_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),filter2(A,P),Xs) )
    <=> ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X5) ) ) ).

% empty_filter_conv
tff(fact_7755_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) )
    <=> ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X5) ) ) ).

% filter_empty_conv
tff(fact_7756_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),insert(A,Y,bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_zw(A,fun(A,$o),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_7757_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,U),U4)
     => ( ! [I3: nat] :
            ( ( aa(nat,$o,ord_less_eq(nat,U),I3)
              & aa(nat,$o,ord_less(nat,I3),U4) )
           => ~ aa(nat,$o,P,I3) )
       => ( aa(list(nat),list(nat),filter2(nat,P),upt(zero_zero(nat),U)) = aa(list(nat),list(nat),filter2(nat,P),upt(zero_zero(nat),U4)) ) ) ) ).

% upt_filter_extend
tff(fact_7758_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : linorder_sort_key(A,A,aTP_Lamp_mp(A,A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),linorder_sort_key(A,A,aTP_Lamp_mp(A,A),aa(list(A),list(A),filter2(A,aTP_Lamp_zx(list(A),fun(A,$o),Xs)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,aTP_Lamp_zy(list(A),fun(A,$o),Xs)),Xs)),linorder_sort_key(A,A,aTP_Lamp_mp(A,A),aa(list(A),list(A),filter2(A,ord_less(A,aa(nat,A,nth(A,Xs),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_7759_sort__key__by__quicksort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : linorder_sort_key(A,B,F2,Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),linorder_sort_key(A,B,F2,aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_zz(fun(A,B),fun(list(A),fun(A,$o)),F2),Xs)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_aaa(fun(A,B),fun(list(A),fun(A,$o)),F2),Xs)),Xs)),linorder_sort_key(A,B,F2,aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_aab(fun(A,B),fun(list(A),fun(A,$o)),F2),Xs)),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_7760_sort__upt,axiom,
    ! [M: nat,Nb: nat] : linorder_sort_key(nat,nat,aTP_Lamp_bg(nat,nat),upt(M,Nb)) = upt(M,Nb) ).

% sort_upt
tff(fact_7761_sort__upto,axiom,
    ! [I: int,J: int] : linorder_sort_key(int,int,aTP_Lamp_bp(int,int),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_7762_sort__key__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B)] : linorder_sort_key(A,B,F2,nil(A)) = nil(A) ) ).

% sort_key_simps(1)
tff(fact_7763_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),linorder_sort_key(A,B,F2,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% length_sort
tff(fact_7764_set__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : aa(list(A),set(A),set2(A),linorder_sort_key(A,B,F2,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_sort
tff(fact_7765_filter__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),linorder_sort_key(A,B,F2,Xs)) = linorder_sort_key(A,B,F2,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% filter_sort
tff(fact_7766_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_aac(fun(A,B),fun(B,fun(A,$o)),F2),K)),linorder_sort_key(A,B,F2,Xs)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_aac(fun(A,B),fun(B,fun(A,$o)),F2),K)),Xs) ) ).

% sort_key_stable
tff(fact_7767_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : linorder_sort_key(A,B,aTP_Lamp_aad(B,fun(A,B),C2),Xs) = Xs ) ).

% sort_key_const
tff(fact_7768_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_aae(list(B),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_aaf(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_aag(list(B),$o)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7769_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_aah(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aai(nat,fun(list(A),$o),I)),Xs)) ) ) ).

% nth_transpose
tff(fact_7770_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_zj(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aaj(list(A),$o)),Xs)) ).

% transpose_max_length
tff(fact_7771_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F2,Xs) = aa(list(B),list(A),map(B,A,comp(option(A),A,B,the2(A),F2)),aa(list(B),list(B),filter2(B,aTP_Lamp_aak(fun(B,option(A)),fun(B,$o),F2)),Xs)) ).

% map_filter_def
tff(fact_7772_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_aal(A,fun(list(A),A))),aa(list(A),list(A),filter2(A,P),linorder_sort_key(A,A,aTP_Lamp_mp(A,A),Xs))) ) ).

% Bleast_code
tff(fact_7773_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xd: B,Xs: list(B)] : map_filter(B,A,F2,aa(list(B),list(B),cons(B,Xd),Xs)) = case_option(list(A),A,map_filter(B,A,F2,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_aam(fun(B,option(A)),fun(list(B),fun(A,list(A))),F2),Xs),aa(B,option(A),F2,Xd)) ).

% map_filter_simps(1)
tff(fact_7774_map__filter__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,option(A))] : map_filter(B,A,F2,nil(B)) = nil(A) ).

% map_filter_simps(2)
tff(fact_7775_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S3: set(A),P: fun(A,$o)] : bleast(A,S3,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aan(set(A),fun(fun(A,$o),fun(A,$o)),S3),P)) ) ).

% Bleast_def
tff(fact_7776_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S3: set(A),P: fun(A,$o)] : abort_Bleast(A,S3,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aan(set(A),fun(fun(A,$o),fun(A,$o)),S3),P)) ) ).

% abort_Bleast_def
tff(fact_7777_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o),Xs: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_aao(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F2),P),Xs) ).

% map_filter_map_filter
tff(fact_7778_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_zo(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aaj(list(A),$o)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_7779_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_zn(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aaj(list(A),$o)),Xss)) ).

% transpose_aux_filter_head
tff(fact_7780_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_7781_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_7782_hd__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_append2
tff(fact_7783_tl__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),Ys) ) ) ).

% tl_append2
tff(fact_7784_hd__replicate,axiom,
    ! [A: $tType,Nb: nat,Xd: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),replicate(A,Nb,Xd)) = Xd ) ) ).

% hd_replicate
tff(fact_7785_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_7786_hd__Cons__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),Xs)),aa(list(A),list(A),tl(A),Xs)) = Xs ) ) ).

% hd_Cons_tl
tff(fact_7787_list_Ocollapse,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) = List ) ) ).

% list.collapse
tff(fact_7788_tl__replicate,axiom,
    ! [A: $tType,Nb: nat,Xd: A] : aa(list(A),list(A),tl(A),replicate(A,Nb,Xd)) = replicate(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xd) ).

% tl_replicate
tff(fact_7789_map__tl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),tl(B),Xs)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),map(B,A,F2),Xs)) ).

% map_tl
tff(fact_7790_list_Omap__sel_I2_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F2: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),list(B),tl(B),aa(list(A),list(B),map(A,B,F2),A2)) = aa(list(A),list(B),map(A,B,F2),aa(list(A),list(A),tl(A),A2)) ) ) ).

% list.map_sel(2)
tff(fact_7791_list_Omap__sel_I1_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F2: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F2),A2)) = aa(A,B,F2,aa(list(A),A,hd(A),A2)) ) ) ).

% list.map_sel(1)
tff(fact_7792_hd__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F2),Xs)) = aa(A,B,F2,aa(list(A),A,hd(A),Xs)) ) ) ).

% hd_map
tff(fact_7793_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(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_aap(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_7794_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_aaq(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_7795_list_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F12: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      aa(list(B),A,case_list(A,B,F12,F22),List) = $ite(List = nil(B),F12,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ).

% list.case_eq_if
tff(fact_7796_list_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F12: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P,aa(list(B),A,case_list(A,B,F12,F22),List))
    <=> ( ( ( List = nil(B) )
         => aa(A,$o,P,F12) )
        & ( ( List = aa(list(B),list(B),cons(B,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
         => aa(A,$o,P,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel
tff(fact_7797_list_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F12: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P,aa(list(B),A,case_list(A,B,F12,F22),List))
    <=> ~ ( ( ( List = nil(B) )
            & ~ aa(A,$o,P,F12) )
          | ( ( List = aa(list(B),list(B),cons(B,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
            & ~ aa(A,$o,P,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel_asm
tff(fact_7798_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_aar(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_7799_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_7800_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_7801_hd__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(Xs = nil(A),aa(list(A),A,hd(A),Ys),aa(list(A),A,hd(A),Xs)) ).

% hd_append
tff(fact_7802_longest__common__prefix,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
    ? [Ps2: list(A),Xs6: list(A),Ys6: list(A)] :
      ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Xs6) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Ys6) )
      & ( ( Xs6 = nil(A) )
        | ( Ys6 = nil(A) )
        | ( aa(list(A),A,hd(A),Xs6) != aa(list(A),A,hd(A),Ys6) ) ) ) ).

% longest_common_prefix
tff(fact_7803_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_7804_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_7805_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list(A),Xd: A] :
      ( ( A2 != nil(A) )
     => ( member(A,Xd,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A2)))
       => member(A,Xd,aa(list(A),set(A),set2(A),A2)) ) ) ).

% list.set_sel(2)
tff(fact_7806_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] :
      aa(list(A),nat,size_list(A,F2),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,F2,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),tl(A),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_7807_hd__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( Xs != nil(list(A)) )
     => ( ( aa(list(list(A)),list(A),hd(list(A)),Xs) != nil(A) )
       => ( aa(list(A),A,hd(A),concat(A,Xs)) = aa(list(A),A,hd(A),aa(list(list(A)),list(A),hd(list(A)),Xs)) ) ) ) ).

% hd_concat
tff(fact_7808_list_Oexpand,axiom,
    ! [A: $tType,List: list(A),List2: list(A)] :
      ( ( ( List = nil(A) )
      <=> ( List2 = nil(A) ) )
     => ( ( ( List != nil(A) )
         => ( ( List2 != nil(A) )
           => ( ( aa(list(A),A,hd(A),List) = aa(list(A),A,hd(A),List2) )
              & ( aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),tl(A),List2) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
tff(fact_7809_list_Osel_I2_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).

% list.sel(2)
tff(fact_7810_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) )
     => ( ! [N: nat] :
            ( aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(B,fun(A,$o),P,aa(nat,B,nth(B,Ys),N)),aa(nat,A,nth(A,Xs),N)) )
       => ! [N9: nat] :
            ( aa(nat,$o,ord_less(nat,N9),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)),N9)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N9)) ) ) ) ).

% list_take_induct_tl2
tff(fact_7811_list_Osel_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),cons(A,X21),X222)) = X21 ).

% list.sel(1)
tff(fact_7812_list_Osel_I3_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),cons(A,X21),X222)) = X222 ).

% list.sel(3)
tff(fact_7813_tl__Nil,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ? [X5: A] : Xs = aa(list(A),list(A),cons(A,X5),nil(A)) ) ) ).

% tl_Nil
tff(fact_7814_Nil__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),tl(A),Xs) )
    <=> ( ( Xs = nil(A) )
        | ? [X5: A] : Xs = aa(list(A),list(A),cons(A,X5),nil(A)) ) ) ).

% Nil_tl
tff(fact_7815_list_Oexhaust__sel,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( List = aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ).

% list.exhaust_sel
tff(fact_7816_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_7817_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_7818_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,From),To)
     => ( aa(nat,$o,ord_less_eq(nat,To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),A,hd(A),slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).

% slice_head
tff(fact_7819_part__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Pivot: B,Xs: list(A)] : linorder_part(A,B,F2,Pivot,Xs) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),product_Pair(list(A),product_prod(list(A),list(A)),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_aas(fun(A,B),fun(B,fun(A,$o)),F2),Pivot)),Xs)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_aac(fun(A,B),fun(B,fun(A,$o)),F2),Pivot)),Xs)),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_aat(fun(A,B),fun(B,fun(A,$o)),F2),Pivot)),Xs))) ) ).

% part_def
tff(fact_7820_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = aa(real,real,sqrt,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% csqrt.simps(1)
tff(fact_7821_complex__Re__numeral,axiom,
    ! [V: num] : re(aa(num,complex,numeral_numeral(complex),V)) = aa(num,real,numeral_numeral(real),V) ).

% complex_Re_numeral
tff(fact_7822_Re__divide__of__nat,axiom,
    ! [Z: complex,Nb: nat] : re(divide_divide(complex,Z,aa(nat,complex,semiring_1_of_nat(complex),Nb))) = divide_divide(real,re(Z),aa(nat,real,semiring_1_of_nat(real),Nb)) ).

% Re_divide_of_nat
tff(fact_7823_Re__divide__of__real,axiom,
    ! [Z: complex,R2: real] : re(divide_divide(complex,Z,real_Vector_of_real(complex,R2))) = divide_divide(real,re(Z),R2) ).

% Re_divide_of_real
tff(fact_7824_Re__sgn,axiom,
    ! [Z: complex] : re(sgn_sgn(complex,Z)) = divide_divide(real,re(Z),real_V7770717601297561774m_norm(complex,Z)) ).

% Re_sgn
tff(fact_7825_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : re(divide_divide(complex,Z,aa(num,complex,numeral_numeral(complex),W))) = divide_divide(real,re(Z),aa(num,real,numeral_numeral(real),W)) ).

% Re_divide_numeral
tff(fact_7826_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero(complex) )
     => ( ( re(Y) = zero_zero(real) )
       => ( cos(real,arg(Y)) = zero_zero(real) ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_7827_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xd: complex] : re(aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),Xd)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),re(Xd)) ).

% scaleR_complex.simps(1)
tff(fact_7828_one__complex_Osimps_I1_J,axiom,
    re(one_one(complex)) = one_one(real) ).

% one_complex.simps(1)
tff(fact_7829_imaginary__unit_Osimps_I1_J,axiom,
    re(imaginary_unit) = zero_zero(real) ).

% imaginary_unit.simps(1)
tff(fact_7830_zero__complex_Osimps_I1_J,axiom,
    re(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(1)
tff(fact_7831_Re__csqrt,axiom,
    ! [Z: complex] : aa(real,$o,ord_less_eq(real,zero_zero(real)),re(csqrt(Z))) ).

% Re_csqrt
tff(fact_7832_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),zero_zero(real))
    <=> ( re(Z) = aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_7833_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A2: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A2)) = re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_7834_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Pivot: B,Xd: A,Xs: list(A)] : linorder_part(A,B,F2,Pivot,aa(list(A),list(A),cons(A,Xd),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_aav(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))))))),F2),Pivot),Xd)),linorder_part(A,B,F2,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_7835_csqrt_Ocode,axiom,
    ! [Z: complex] :
      csqrt(Z) = complex2(aa(real,real,sqrt,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),bit0(one2)))),
        aa(real,real,
          aa(real,fun(real,real),times_times(real),
            $ite(im(Z) = zero_zero(real),one_one(real),sgn_sgn(real,im(Z)))),
          aa(real,real,sqrt,divide_divide(real,aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% csqrt.code
tff(fact_7836_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      im(csqrt(Z)) = aa(real,real,
        aa(real,fun(real,real),times_times(real),
          $ite(im(Z) = zero_zero(real),one_one(real),sgn_sgn(real,im(Z)))),
        aa(real,real,sqrt,divide_divide(real,aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt.simps(2)
tff(fact_7837_complex__Im__fact,axiom,
    ! [Nb: nat] : im(semiring_char_0_fact(complex,Nb)) = zero_zero(real) ).

% complex_Im_fact
tff(fact_7838_complex__Im__of__int,axiom,
    ! [Z: int] : im(aa(int,complex,ring_1_of_int(complex),Z)) = zero_zero(real) ).

% complex_Im_of_int
tff(fact_7839_Im__complex__of__real,axiom,
    ! [Z: real] : im(real_Vector_of_real(complex,Z)) = zero_zero(real) ).

% Im_complex_of_real
tff(fact_7840_Im__power__real,axiom,
    ! [Xd: complex,Nb: nat] :
      ( ( im(Xd) = zero_zero(real) )
     => ( im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xd),Nb)) = zero_zero(real) ) ) ).

% Im_power_real
tff(fact_7841_complex__Im__numeral,axiom,
    ! [V: num] : im(aa(num,complex,numeral_numeral(complex),V)) = zero_zero(real) ).

% complex_Im_numeral
tff(fact_7842_complex__Im__of__nat,axiom,
    ! [Nb: nat] : im(aa(nat,complex,semiring_1_of_nat(complex),Nb)) = zero_zero(real) ).

% complex_Im_of_nat
tff(fact_7843_Im__divide__of__real,axiom,
    ! [Z: complex,R2: real] : im(divide_divide(complex,Z,real_Vector_of_real(complex,R2))) = divide_divide(real,im(Z),R2) ).

% Im_divide_of_real
tff(fact_7844_Im__sgn,axiom,
    ! [Z: complex] : im(sgn_sgn(complex,Z)) = divide_divide(real,im(Z),real_V7770717601297561774m_norm(complex,Z)) ).

% Im_sgn
tff(fact_7845_Re__power__real,axiom,
    ! [Xd: complex,Nb: nat] :
      ( ( im(Xd) = zero_zero(real) )
     => ( re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xd),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xd)),Nb) ) ) ).

% Re_power_real
tff(fact_7846_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : im(divide_divide(complex,Z,aa(num,complex,numeral_numeral(complex),W))) = divide_divide(real,im(Z),aa(num,real,numeral_numeral(real),W)) ).

% Im_divide_numeral
tff(fact_7847_Im__divide__of__nat,axiom,
    ! [Z: complex,Nb: nat] : im(divide_divide(complex,Z,aa(nat,complex,semiring_1_of_nat(complex),Nb))) = divide_divide(real,im(Z),aa(nat,real,semiring_1_of_nat(real),Nb)) ).

% Im_divide_of_nat
tff(fact_7848_csqrt__of__real__nonneg,axiom,
    ! [Xd: complex] :
      ( ( im(Xd) = zero_zero(real) )
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),re(Xd))
       => ( csqrt(Xd) = real_Vector_of_real(complex,aa(real,real,sqrt,re(Xd))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_7849_csqrt__minus,axiom,
    ! [Xd: complex] :
      ( ( aa(real,$o,ord_less(real,im(Xd)),zero_zero(real))
        | ( ( im(Xd) = zero_zero(real) )
          & aa(real,$o,ord_less_eq(real,zero_zero(real)),re(Xd)) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),Xd)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(Xd)) ) ) ).

% csqrt_minus
tff(fact_7850_csqrt__of__real__nonpos,axiom,
    ! [Xd: complex] :
      ( ( im(Xd) = zero_zero(real) )
     => ( aa(real,$o,ord_less_eq(real,re(Xd)),zero_zero(real))
       => ( csqrt(Xd) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,aa(real,real,sqrt,abs_abs(real,re(Xd))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_7851_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( member(complex,Z,ring_1_Ints(complex))
    <=> ( ( im(Z) = zero_zero(real) )
        & ? [I2: int] : re(Z) = aa(int,real,ring_1_of_int(real),I2) ) ) ).

% complex_is_Int_iff
tff(fact_7852_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one(real) ).

% imaginary_unit.simps(2)
tff(fact_7853_one__complex_Osimps_I2_J,axiom,
    im(one_one(complex)) = zero_zero(real) ).

% one_complex.simps(2)
tff(fact_7854_zero__complex_Osimps_I2_J,axiom,
    im(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(2)
tff(fact_7855_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xd: complex] : im(aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),Xd)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),im(Xd)) ).

% scaleR_complex.simps(2)
tff(fact_7856_times__complex_Osimps_I2_J,axiom,
    ! [Xd: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xd),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),re(Y))) ).

% times_complex.simps(2)
tff(fact_7857_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( im(Z) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z) = abs_abs(real,re(Z)) ) ) ).

% cmod_eq_Re
tff(fact_7858_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( re(Z) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z) = abs_abs(real,im(Z)) ) ) ).

% cmod_eq_Im
tff(fact_7859_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( abs_abs(real,re(Z)) = real_V7770717601297561774m_norm(complex,Z) )
     => ( im(Z) = zero_zero(real) ) ) ).

% Im_eq_0
tff(fact_7860_times__complex_Osimps_I1_J,axiom,
    ! [Xd: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xd),Y)) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),im(Y))) ).

% times_complex.simps(1)
tff(fact_7861_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xd: complex] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),Xd) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),re(Xd)),aa(real,real,aa(real,fun(real,real),times_times(real),R2),im(Xd))) ).

% scaleR_complex.code
tff(fact_7862_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),re(csqrt(Z)))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & aa(real,$o,ord_less_eq(real,zero_zero(real)),im(csqrt(Z))) ) ) ).

% csqrt_principal
tff(fact_7863_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A2: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A2)) = im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_7864_Re__exp,axiom,
    ! [Z: complex] : re(aa(complex,complex,exp(complex),Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,exp(real),re(Z))),cos(real,im(Z))) ).

% Re_exp
tff(fact_7865_Im__exp,axiom,
    ! [Z: complex] : im(aa(complex,complex,exp(complex),Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,exp(real),re(Z))),sin(real,im(Z))) ).

% Im_exp
tff(fact_7866_times__complex_Ocode,axiom,
    ! [Xd: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xd),Y) = complex2(aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),re(Y)))) ).

% times_complex.code
tff(fact_7867_cmod__power2,axiom,
    ! [Z: complex] : aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),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),re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% cmod_power2
tff(fact_7868_Im__power2,axiom,
    ! [Xd: complex] : im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xd),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),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(Xd))),im(Xd)) ).

% Im_power2
tff(fact_7869_Re__power2,axiom,
    ! [Xd: complex] : re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xd),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Re_power2
tff(fact_7870_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero(complex) )
    <=> ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(real) ) ) ).

% complex_eq_0
tff(fact_7871_norm__complex__def,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,Z) = 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),re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% norm_complex_def
tff(fact_7872_inverse__complex_Osimps_I1_J,axiom,
    ! [Xd: complex] : re(aa(complex,complex,inverse_inverse(complex),Xd)) = divide_divide(real,re(Xd),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% inverse_complex.simps(1)
tff(fact_7873_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
    <=> aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% complex_neq_0
tff(fact_7874_Re__divide,axiom,
    ! [Xd: complex,Y: complex] : re(divide_divide(complex,Xd,Y)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Re_divide
tff(fact_7875_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( aa(real,$o,ord_less(real,zero_zero(real)),re(B2))
        | ( ( re(B2) = zero_zero(real) )
          & aa(real,$o,ord_less_eq(real,zero_zero(real)),im(B2)) ) )
     => ( csqrt(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = B2 ) ) ).

% csqrt_square
tff(fact_7876_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),W),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z )
     => ( ( aa(real,$o,ord_less(real,zero_zero(real)),re(W))
          | ( ( re(W) = zero_zero(real) )
            & aa(real,$o,ord_less_eq(real,zero_zero(real)),im(W)) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_7877_inverse__complex_Osimps_I2_J,axiom,
    ! [Xd: complex] : im(aa(complex,complex,inverse_inverse(complex),Xd)) = divide_divide(real,aa(real,real,uminus_uminus(real),im(Xd)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% inverse_complex.simps(2)
tff(fact_7878_Im__divide,axiom,
    ! [Xd: complex,Y: complex] : im(divide_divide(complex,Xd,Y)) = divide_divide(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Im_divide
tff(fact_7879_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),abs_abs(real,re(Z))),abs_abs(real,im(Z)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),real_V7770717601297561774m_norm(complex,Z))) ).

% complex_abs_le_norm
tff(fact_7880_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,re(Z),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,im(Z),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) ) ) ).

% complex_unit_circle
tff(fact_7881_inverse__complex_Ocode,axiom,
    ! [Xd: complex] : aa(complex,complex,inverse_inverse(complex),Xd) = complex2(divide_divide(real,re(Xd),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),divide_divide(real,aa(real,real,uminus_uminus(real),im(Xd)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xd)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% inverse_complex.code
tff(fact_7882_Complex__divide,axiom,
    ! [Xd: complex,Y: complex] : divide_divide(complex,Xd,Y) = complex2(divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),divide_divide(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(Xd)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xd)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% Complex_divide
tff(fact_7883_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member(complex,R2,real_Vector_Reals(complex))
     => ( im(divide_divide(complex,R2,Z)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R2))),im(Z)),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Im_Reals_divide
tff(fact_7884_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member(complex,R2,real_Vector_Reals(complex))
     => ( re(divide_divide(complex,R2,Z)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(R2)),re(Z)),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Re_Reals_divide
tff(fact_7885_Re__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( member(complex,R2,real_Vector_Reals(complex))
     => ( re(divide_divide(complex,Z,R2)) = divide_divide(real,re(Z),re(R2)) ) ) ).

% Re_divide_Reals
tff(fact_7886_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xd: complex] :
      ( member(complex,Y,real_Vector_Reals(complex))
     => ( member(complex,Xd,real_Vector_Reals(complex))
       => ( ( Xd = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) )
        <=> ( ( Xd = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_7887_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xd: complex] :
      ( member(complex,Y,real_Vector_Reals(complex))
     => ( member(complex,Xd,real_Vector_Reals(complex))
       => ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) = Xd )
        <=> ( ( Xd = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_7888_Im__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( member(complex,R2,real_Vector_Reals(complex))
     => ( im(divide_divide(complex,Z,R2)) = divide_divide(real,im(Z),re(R2)) ) ) ).

% Im_divide_Reals
tff(fact_7889_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( member(complex,Z,real_Vector_Reals(complex))
    <=> ( im(Z) = zero_zero(real) ) ) ).

% complex_is_Real_iff
tff(fact_7890_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => ( ( B2 != zero_zero(A) )
             => member(A,divide_divide(A,A2,B2),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_7891_Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,divide_divide(A,A2,B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_divide
tff(fact_7892_Reals__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_add
tff(fact_7893_Reals__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,Nb: nat] :
          ( member(A,A2,real_Vector_Reals(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),real_Vector_Reals(A)) ) ) ).

% Reals_power
tff(fact_7894_Reals__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_mult
tff(fact_7895_Complex__in__Reals,axiom,
    ! [Xd: real] : member(complex,complex2(Xd,zero_zero(real)),real_Vector_Reals(complex)) ).

% Complex_in_Reals
tff(fact_7896_Reals__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,zero_zero(A),real_Vector_Reals(A)) ) ).

% Reals_0
tff(fact_7897_Reals__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,one_one(A),real_Vector_Reals(A)) ) ).

% Reals_1
tff(fact_7898_Reals__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : member(A,aa(num,A,numeral_numeral(A),W),real_Vector_Reals(A)) ) ).

% Reals_numeral
tff(fact_7899_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( ( A2 != zero_zero(A) )
           => member(A,aa(A,A,inverse_inverse(A),A2),real_Vector_Reals(A)) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_7900_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N2: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N: nat] : member(complex,aa(nat,complex,G,N),real_Vector_Reals(complex))
           => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),re(aa(nat,complex,G,N)))
             => ( ! [N: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,N2),N)
                   => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G,N))) )
               => summable(A,F2) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_7901_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs3: 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))),product_Pair(list(A),product_prod(A,list(A)),Ys),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs3))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),Zs3)) )
        & aa(A,$o,P,Y)
        & ~ ? [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X4) ) ) ) ).

% extract_SomeE
tff(fact_7902_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs3: 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))),product_Pair(list(A),product_prod(A,list(A)),Ys),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs3))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),Zs3)) )
        & aa(A,$o,P,Y)
        & ~ ? [X5: A] :
              ( member(A,X5,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X5) ) ) ) ).

% extract_Some_iff
tff(fact_7903_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_7904_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)))) )
    <=> ~ ? [X5: A] :
            ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X5) ) ) ).

% extract_None_iff
tff(fact_7905_complex__mult__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_mult_cnj
tff(fact_7906_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3))
          <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aaw(set(A),fun(fun(A,B),fun(A,$o)),A3),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_7907_card__UNIV__unit,axiom,
    finite_card(product_unit,top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_7908_card__Collect__less__nat,axiom,
    ! [Nb: nat] : finite_card(nat,collect(nat,aTP_Lamp_as(nat,fun(nat,$o),Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_7909_card__atMost,axiom,
    ! [U: nat] : finite_card(nat,set_ord_atMost(nat,U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_7910_complex__cnj__zero,axiom,
    cnj(zero_zero(complex)) = zero_zero(complex) ).

% complex_cnj_zero
tff(fact_7911_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% complex_cnj_zero_iff
tff(fact_7912_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% complex_cnj_one_iff
tff(fact_7913_complex__cnj__one,axiom,
    cnj(one_one(complex)) = one_one(complex) ).

% complex_cnj_one
tff(fact_7914_card__Collect__le__nat,axiom,
    ! [Nb: nat] : finite_card(nat,collect(nat,aTP_Lamp_aax(nat,fun(nat,$o),Nb))) = aa(nat,nat,suc,Nb) ).

% card_Collect_le_nat
tff(fact_7915_card__UNIV__bool,axiom,
    finite_card($o,top_top(set($o))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% card_UNIV_bool
tff(fact_7916_card_Oempty,axiom,
    ! [A: $tType] : finite_card(A,bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_7917_card_Oinfinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( finite_card(A,A3) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_7918_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card(nat,set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_7919_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(list(A),nat,size_size(list(A)),linord4507533701916653071of_set(A,A3)) = finite_card(A,A3) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_7920_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A3: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_aay(A,fun(B,A),Y),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),finite_card(B,A3)) ) ).

% prod_constant
tff(fact_7921_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card(nat,set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,minus_minus(nat,U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_7922_card__0__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( finite_card(A,A3) = zero_zero(nat) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_7923_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ~ member(A,Xd,A3)
       => ( finite_card(A,insert(A,Xd,A3)) = aa(nat,nat,suc,finite_card(A,A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_7924_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = zero_zero(real) ).

% complex_In_mult_cnj_zero
tff(fact_7925_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_aaz(A,fun(B,A),Y)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,A3))),Y) ) ).

% sum_constant
tff(fact_7926_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( member(A,A2,A3)
     => ( ~ member(A,A2,B4)
       => ( finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,A2,B4))) = aa(nat,nat,minus_minus(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),B4))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_7927_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or1337092689740270186AtMost(int,L,U)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_7928_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or5935395276787703475ssThan(int,L,U)) = nat2(aa(int,int,minus_minus(int,U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_7929_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( finite_card(A,insert(A,A2,insert(A,B2,bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_7930_subset__card__intvl__is__intvl,axiom,
    ! [A3: set(nat),K: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),A3),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),finite_card(nat,A3))))
     => ( A3 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),finite_card(nat,A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_7931_card__map__elide,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Nb),finite_card(word(A),top_top(set(word(A)))))
         => ( finite_card(word(A),image(nat,word(A),semiring_1_of_nat(word(A)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) = finite_card(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ) ).

% card_map_elide
tff(fact_7932_card__less__Suc2,axiom,
    ! [M7: set(nat),I: nat] :
      ( ~ member(nat,zero_zero(nat),M7)
     => ( finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aba(set(nat),fun(nat,fun(nat,$o)),M7),I))) = finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_abb(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc2
tff(fact_7933_card__less__Suc,axiom,
    ! [M7: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M7)
     => ( aa(nat,nat,suc,finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aba(set(nat),fun(nat,fun(nat,$o)),M7),I)))) = finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_abb(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc
tff(fact_7934_card__less,axiom,
    ! [M7: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M7)
     => ( finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_abb(set(nat),fun(nat,fun(nat,$o)),M7),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_7935_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_am(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),finite_card(A,A3)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_7936_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,ord_less_eq(nat,finite_card(A,aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% card_length
tff(fact_7937_card__eq__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( finite_card(A,A3) = zero_zero(nat) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% card_eq_0_iff
tff(fact_7938_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( finite_card(A,A3) = one_one(nat) )
     => ~ ! [X: A] : A3 != insert(A,X,bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_7939_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( finite_card(A,A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B11: set(A)] :
          ( ( A3 = insert(A,B7,B11) )
          & ~ member(A,B7,B11)
          & ( finite_card(A,B11) = K )
          & aa(set(A),$o,finite_finite2(A),B11) ) ) ).

% card_Suc_eq_finite
tff(fact_7940_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( finite_card(A,insert(A,Xd,A3)) = $ite(member(A,Xd,A3),finite_card(A,A3),aa(nat,nat,suc,finite_card(A,A3))) ) ) ).

% card_insert_if
tff(fact_7941_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(nat,$o,ord_less(nat,finite_card(A,image(B,A,F2,A3))),finite_card(B,A3))
     => ~ inj_on(B,A,F2,A3) ) ).

% pigeonhole
tff(fact_7942_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,A3))
     => aa(set(A),$o,finite_finite2(A),A3) ) ).

% card_ge_0_finite
tff(fact_7943_card__less__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),B4)
       => ( aa(nat,$o,ord_less(nat,finite_card(A,A3)),finite_card(A,B4))
         => aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),B4))),finite_card(A,aa(set(A),set(A),minus_minus(set(A),B4),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_7944_psubset__card__mono,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,ord_less(set(A),A3),B4)
       => aa(nat,$o,ord_less(nat,finite_card(A,A3)),finite_card(A,B4)) ) ) ).

% psubset_card_mono
tff(fact_7945_card__2__iff_H,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( finite_card(A,S3) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X5: A] :
          ( member(A,X5,S3)
          & ? [Xa3: A] :
              ( member(A,Xa3,S3)
              & ( X5 != Xa3 )
              & ! [Xb3: A] :
                  ( member(A,Xb3,S3)
                 => ( ( Xb3 = X5 )
                    | ( Xb3 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_7946_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : finite_card(A,A3) = finite_fold(A,nat,aTP_Lamp_ax(A,fun(nat,nat)),zero_zero(nat),A3) ).

% card.eq_fold
tff(fact_7947_card__eq__sum,axiom,
    ! [A: $tType,A3: set(A)] : finite_card(A,A3) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_abc(A,nat)),A3) ).

% card_eq_sum
tff(fact_7948_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card(int,set_or7035219750837199246ssThan(int,zero_zero(int),U)) = nat2(U) ).

% card_atLeastZeroLessThan_int
tff(fact_7949_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_abd(fun(A,nat),fun(A,nat),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),finite_card(A,A3)) ).

% sum_Suc
tff(fact_7950_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S3: set(A),T4: set(B),R: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( aa(set(B),$o,finite_finite2(B),T4)
       => ( ! [X: B] :
              ( member(B,X,T4)
             => ( finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_abe(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S3),R),X))) = K ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_abg(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T4),R)),S3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),finite_card(B,T4)) ) ) ) ) ).

% sum_multicount
tff(fact_7951_real__of__card,axiom,
    ! [A: $tType,A3: set(A)] : aa(nat,real,semiring_1_of_nat(real),finite_card(A,A3)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_abh(A,real)),A3) ).

% real_of_card
tff(fact_7952_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I3: A] :
              ( member(A,I3,A3)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),K6) )
         => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3))),K6)) ) ) ).

% sum_bounded_above
tff(fact_7953_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),K6: B,F2: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A3)
             => aa(B,$o,ord_less_eq(B,K6),aa(A,B,F2,I3)) )
         => aa(B,$o,ord_less_eq(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3))),K6)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ).

% sum_bounded_below
tff(fact_7954_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,A3))
    <=> ( ( A3 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% card_gt_0_iff
tff(fact_7955_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_7956_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,ord_less_eq(nat,finite_card(A,A3)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X5: A] :
            ( member(A,X5,A3)
           => ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7957_card__1__singleton__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( finite_card(A,A3) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X5: A] : A3 = insert(A,X5,bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_7958_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( finite_card(A,A3) = aa(nat,nat,suc,K) )
     => ? [B3: A,B9: set(A)] :
          ( ( A3 = insert(A,B3,B9) )
          & ~ member(A,B3,B9)
          & ( finite_card(A,B9) = K )
          & ( ( K = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_7959_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( finite_card(A,A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B11: set(A)] :
          ( ( A3 = insert(A,B7,B11) )
          & ~ member(A,B7,B11)
          & ( finite_card(A,B11) = K )
          & ( ( K = zero_zero(nat) )
           => ( B11 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_7960_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A3: set(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Nb)),finite_card(A,A3))
    <=> ? [A7: A,B11: set(A)] :
          ( ( A3 = insert(A,A7,B11) )
          & ~ member(A,A7,B11)
          & aa(nat,$o,ord_less_eq(nat,Nb),finite_card(A,B11))
          & aa(set(A),$o,finite_finite2(A),B11) ) ) ).

% card_le_Suc_iff
tff(fact_7961_card__1__singletonI,axiom,
    ! [A: $tType,S3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( ( finite_card(A,S3) = one_one(nat) )
       => ( member(A,Xd,S3)
         => ( S3 = insert(A,Xd,bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_7962_card__psubset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,ord_less_eq(set(A),A3),B4)
       => ( aa(nat,$o,ord_less(nat,finite_card(A,A3)),finite_card(A,B4))
         => aa(set(A),$o,ord_less(set(A),A3),B4) ) ) ) ).

% card_psubset
tff(fact_7963_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_an(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),finite_card(A,A3))),set_ord_atMost(nat,Nb)) ) ) ).

% card_lists_length_le
tff(fact_7964_Re__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( re(divide_divide(complex,A2,B2)) = zero_zero(real) )
    <=> ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Re_complex_div_eq_0
tff(fact_7965_Im__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( im(divide_divide(complex,A2,B2)) = zero_zero(real) )
    <=> ( im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Im_complex_div_eq_0
tff(fact_7966_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M7: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M7)
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(A,M7)),M7) ) ).

% ex_bij_betw_nat_finite
tff(fact_7967_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M7: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M7)
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or1337092689740270186AtMost(nat,one_one(nat),finite_card(A,M7)),M7) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_7968_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Nb)
         => aa(nat,$o,ord_less_eq(nat,finite_card(A,collect(A,aTP_Lamp_al(nat,fun(A,$o),Nb)))),Nb) ) ) ).

% card_roots_unity
tff(fact_7969_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M7: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M7)
     => ? [H3: fun(A,nat)] : bij_betw(A,nat,H3,M7,set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(A,M7))) ) ).

% ex_bij_betw_finite_nat
tff(fact_7970_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(nat,$o,ord_less_eq(nat,finite_card(nat,N2)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_7971_card__le__Suc__Max,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S3)
     => aa(nat,$o,ord_less_eq(nat,finite_card(nat,S3)),aa(nat,nat,suc,lattic643756798349783984er_Max(nat,S3))) ) ).

% card_le_Suc_Max
tff(fact_7972_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P3),Xs)) = finite_card(nat,collect(nat,aa(list(A),fun(nat,$o),aTP_Lamp_abi(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_7973_card__UNIV__char,axiom,
    finite_card(char,top_top(set(char))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))) ).

% card_UNIV_char
tff(fact_7974_card__sum__le__nat__sum,axiom,
    ! [S3: set(nat)] : aa(nat,$o,ord_less_eq(nat,aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_bg(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(nat,S3)))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_bg(nat,nat)),S3)) ).

% card_sum_le_nat_sum
tff(fact_7975_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( finite_card(complex,collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_kr(complex,fun(nat,fun(complex,$o)),C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_7976_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( finite_card(complex,collect(complex,aTP_Lamp_gb(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_7977_card__2__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( finite_card(A,S3) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X5: A,Y5: A] :
          ( ( S3 = insert(A,X5,insert(A,Y5,bot_bot(set(A)))) )
          & ( X5 != Y5 ) ) ) ).

% card_2_iff
tff(fact_7978_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),image(B,A,F2,top_top(set(B))))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,image(B,A,F2,top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_7979_card__3__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( finite_card(A,S3) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X5: A,Y5: A,Z5: A] :
          ( ( S3 = insert(A,X5,insert(A,Y5,insert(A,Z5,bot_bot(set(A))))) )
          & ( X5 != Y5 )
          & ( Y5 != Z5 )
          & ( X5 != Z5 ) ) ) ).

% card_3_iff
tff(fact_7980_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,A3))
     => ( A3 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_7981_card__insert__disjoint_H,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ~ member(A,Xd,A3)
       => ( aa(nat,nat,minus_minus(nat,finite_card(A,insert(A,Xd,A3))),aa(nat,nat,suc,zero_zero(nat))) = finite_card(A,A3) ) ) ) ).

% card_insert_disjoint'
tff(fact_7982_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xd,A3)
       => ( aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))) = finite_card(A,A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_7983_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( finite_card(A,insert(A,Xd,A3)) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_7984_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xd,A3)
       => ( finite_card(A,A3) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_7985_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xd,A3)
       => aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))),finite_card(A,A3)) ) ) ).

% card_Diff1_less
tff(fact_7986_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),Xd: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xd,A3)
       => ( member(A,Y,A3)
         => aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A))))),insert(A,Y,bot_bot(set(A)))))),finite_card(A,A3)) ) ) ) ).

% card_Diff2_less
tff(fact_7987_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      ( aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A)))))),finite_card(A,A3))
    <=> ( aa(set(A),$o,finite_finite2(A),A3)
        & member(A,Xd,A3) ) ) ).

% card_Diff1_less_iff
tff(fact_7988_card__Diff__singleton__if,axiom,
    ! [A: $tType,A3: set(A),Xd: A] :
      finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A))))) = $ite(member(A,Xd,A3),aa(nat,nat,minus_minus(nat,finite_card(A,A3)),one_one(nat)),finite_card(A,A3)) ).

% card_Diff_singleton_if
tff(fact_7989_card__Diff__singleton,axiom,
    ! [A: $tType,Xd: A,A3: set(A)] :
      ( member(A,Xd,A3)
     => ( finite_card(A,aa(set(A),set(A),minus_minus(set(A),A3),insert(A,Xd,bot_bot(set(A))))) = aa(nat,nat,minus_minus(nat,finite_card(A,A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_7990_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less(real,re(divide_divide(complex,A2,B2))),zero_zero(real))
    <=> aa(real,$o,ord_less(real,re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Re_complex_div_lt_0
tff(fact_7991_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),re(divide_divide(complex,A2,B2)))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Re_complex_div_gt_0
tff(fact_7992_Re__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less_eq(real,re(divide_divide(complex,A2,B2))),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Re_complex_div_le_0
tff(fact_7993_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),re(divide_divide(complex,A2,B2)))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Re_complex_div_ge_0
tff(fact_7994_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less(real,im(divide_divide(complex,A2,B2))),zero_zero(real))
    <=> aa(real,$o,ord_less(real,im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Im_complex_div_lt_0
tff(fact_7995_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),im(divide_divide(complex,A2,B2)))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Im_complex_div_gt_0
tff(fact_7996_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S3: set(A),F2: fun(A,B),K6: real] :
          ( ! [X: A] :
              ( member(A,X,S3)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X))),K6) )
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S3))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),finite_card(A,S3))),K6)) ) ) ).

% sum_norm_bound
tff(fact_7997_Im__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less_eq(real,im(divide_divide(complex,A2,B2))),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Im_complex_div_le_0
tff(fact_7998_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),im(divide_divide(complex,A2,B2)))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Im_complex_div_ge_0
tff(fact_7999_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),Nb: B,K: nat] :
          ( ! [I3: A] :
              ( member(A,I3,A3)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3))
                & aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),Nb) ) )
         => ( aa(nat,$o,ord_less_eq(nat,finite_card(A,A3)),K)
           => ( aa(B,$o,ord_less_eq(B,one_one(B)),Nb)
             => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Nb),K)) ) ) ) ) ).

% prod_le_power
tff(fact_8000_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I3: A] :
              ( member(A,I3,A3)
             => aa(B,$o,ord_less(B,aa(A,B,F2,I3)),K6) )
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,A3))
           => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3))),K6)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_8001_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I3: A] :
              ( member(A,I3,A3)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),divide_divide(B,K6,aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3)))) )
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),K6) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8002_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% complex_mod_mult_cnj
tff(fact_8003_card__insert__le__m1,axiom,
    ! [A: $tType,Nb: nat,Y: set(A),Xd: A] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less_eq(nat,finite_card(A,Y)),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))
       => aa(nat,$o,ord_less_eq(nat,finite_card(A,insert(A,Xd,Y))),Nb) ) ) ).

% card_insert_le_m1
tff(fact_8004_complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa(real,$o,ord_less(real,zero_zero(real)),re(divide_divide(complex,A2,B2)))
      <=> aa(real,$o,ord_less(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) )
      & ( aa(real,$o,ord_less(real,zero_zero(real)),im(divide_divide(complex,A2,B2)))
      <=> aa(real,$o,ord_less(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ) ).

% complex_div_gt_0
tff(fact_8005_card__word__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: word(A)] : finite_card(word(A),top_top(set(word(A)))) = 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)),Xd)) ) ).

% card_word_size
tff(fact_8006_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S3: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),R)
           => ( aa(set(B),$o,ord_less_eq(set(B),image(A,B,G,S3)),R)
             => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abj(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_abl(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_8007_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( groups7121269368397514597t_prod(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_abm(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C2),S3) = $ite(member(A,A2,S3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(nat,nat,minus_minus(nat,finite_card(A,S3)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),finite_card(A,S3))) ) ) ) ).

% prod_gen_delta
tff(fact_8008_polyfun__roots__card,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,ord_less_eq(nat,K),Nb)
           => aa(nat,$o,ord_less_eq(nat,finite_card(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_hh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_8009_sum__le__card__Max,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(nat,$o,ord_less_eq(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),finite_card(A,A3)),lattic643756798349783984er_Max(nat,image(A,nat,F2,A3)))) ) ).

% sum_le_card_Max
tff(fact_8010_complex__norm__square,axiom,
    ! [Z: complex] : real_Vector_of_real(complex,aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) ).

% complex_norm_square
tff(fact_8011_card__map__elide2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Nb),finite_card(word(A),top_top(set(word(A)))))
         => ( finite_card(word(A),image(nat,word(A),semiring_1_of_nat(word(A)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) = Nb ) ) ) ).

% card_map_elide2
tff(fact_8012_card__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( finite_card(word(A),top_top(set(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))) ) ) ).

% card_word
tff(fact_8013_complex__add__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Z),cnj(Z)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(Z))) ).

% complex_add_cnj
tff(fact_8014_polyfun__rootbound,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,ord_less_eq(nat,K),Nb)
           => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_hh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
              & aa(nat,$o,ord_less_eq(nat,finite_card(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_hh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_8015_complex__diff__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,minus_minus(complex,Z),cnj(Z)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),im(Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_8016_complex__div__cnj,axiom,
    ! [A2: complex,B2: complex] : divide_divide(complex,A2,B2) = divide_divide(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)),real_Vector_of_real(complex,aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,B2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_div_cnj
tff(fact_8017_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(Z)),W)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))))) ).

% cnj_add_mult_eq_Re
tff(fact_8018_card__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A))))) ) ) ).

% card_bit1
tff(fact_8019_Cardinality_Ocard__set,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(set(A),top_top(set(set(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A)))) ) ) ).

% Cardinality.card_set
tff(fact_8020_card__num0,axiom,
    finite_card(numeral_num0,top_top(set(numeral_num0))) = zero_zero(nat) ).

% card_num0
tff(fact_8021_card__num1,axiom,
    finite_card(numeral_num1,top_top(set(numeral_num1))) = one_one(nat) ).

% card_num1
tff(fact_8022_card__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( finite_finite(B)
        & finite_finite(A) )
     => ( finite_card(sum_sum(A,B),top_top(set(sum_sum(A,B)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,top_top(set(A)))),finite_card(B,top_top(set(B)))) ) ) ).

% card_sum
tff(fact_8023_card__prod,axiom,
    ! [A: $tType,B: $tType] : finite_card(product_prod(A,B),top_top(set(product_prod(A,B)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),finite_card(A,top_top(set(A)))),finite_card(B,top_top(set(B)))) ).

% card_prod
tff(fact_8024_card__nat,axiom,
    finite_card(nat,top_top(set(nat))) = zero_zero(nat) ).

% card_nat
tff(fact_8025_card__option,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(option(A),top_top(set(option(A)))) = aa(nat,nat,suc,finite_card(A,top_top(set(A)))) ) ) ).

% card_option
tff(fact_8026_card__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A)))) ) ) ).

% card_bit0
tff(fact_8027_card__literal,axiom,
    finite_card(literal,top_top(set(literal))) = zero_zero(nat) ).

% card_literal
tff(fact_8028_CARD__1,axiom,
    ! [A: $tType] :
      ( cARD_1(A)
     => ( finite_card(A,top_top(set(A))) = one_one(nat) ) ) ).

% CARD_1
tff(fact_8029_bit1_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size0
tff(fact_8030_bit0_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size0
tff(fact_8031_bit0_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size1
tff(fact_8032_bit1_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size1
tff(fact_8033_zero__less__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,top_top(set(A)))) ) ).

% zero_less_card_finite
tff(fact_8034_card__UNIV__sum,axiom,
    ! [B: $tType,A: $tType] :
      finite_card(sum_sum(A,B),top_top(set(sum_sum(A,B)))) = $ite(
        ( ( finite_card(A,top_top(set(A))) != zero_zero(nat) )
        & ( finite_card(B,top_top(set(B))) != zero_zero(nat) ) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,top_top(set(A)))),finite_card(B,top_top(set(B)))),
        zero_zero(nat) ) ).

% card_UNIV_sum
tff(fact_8035_finite__UNIV__fun,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
    <=> ( ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
          & aa(set(B),$o,finite_finite2(B),top_top(set(B))) )
        | ( finite_card(B,top_top(set(B))) = one_one(nat) ) ) ) ).

% finite_UNIV_fun
tff(fact_8036_bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit0(A)] :
          ~ ! [Z2: int] :
              ( ( Xd = aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z2) )
             => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
               => ~ aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ) ) ).

% bit0_cases
tff(fact_8037_bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A)] :
          ~ ! [Z2: int] :
              ( ( Xd = aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z2) )
             => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
               => ~ aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ) ) ).

% bit1_cases
tff(fact_8038_bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit0(A),$o),Xd: numeral_bit0(A)] :
          ( ! [Z2: int] :
              ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
             => ( aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))
               => aa(numeral_bit0(A),$o,P,aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z2)) ) )
         => aa(numeral_bit0(A),$o,P,Xd) ) ) ).

% bit0_induct
tff(fact_8039_bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),Xd: numeral_bit1(A)] :
          ( ! [Z2: int] :
              ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
             => ( aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))
               => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z2)) ) )
         => aa(numeral_bit1(A),$o,P,Xd) ) ) ).

% bit1_induct
tff(fact_8040_one__less__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),finite_card(A,top_top(set(A)))) ) ).

% one_less_card
tff(fact_8041_one__le__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),finite_card(A,top_top(set(A)))) ) ).

% one_le_card_finite
tff(fact_8042_card__fun,axiom,
    ! [A: $tType,B: $tType] :
      finite_card(fun(A,B),top_top(set(fun(A,B)))) = $ite(
        ( ( ( finite_card(A,top_top(set(A))) != zero_zero(nat) )
          & ( finite_card(B,top_top(set(B))) != zero_zero(nat) ) )
        | ( finite_card(B,top_top(set(B))) = one_one(nat) ) ),
        aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),finite_card(B,top_top(set(B)))),finite_card(A,top_top(set(A)))),
        zero_zero(nat) ) ).

% card_fun
tff(fact_8043_one__less__int__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(int,$o,ord_less(int,one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))) ) ).

% one_less_int_card
tff(fact_8044_two__le__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A)))) ) ).

% two_le_card
tff(fact_8045_card__UNIV__option,axiom,
    ! [A: $tType] :
      finite_card(option(A),top_top(set(option(A)))) = $ite(finite_card(A,top_top(set(A))) = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,top_top(set(A)))),one_one(nat))) ).

% card_UNIV_option
tff(fact_8046_card__UNIV__set,axiom,
    ! [A: $tType] :
      finite_card(set(A),top_top(set(set(A)))) = $ite(finite_card(A,top_top(set(A))) = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A))))) ).

% card_UNIV_set
tff(fact_8047_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => ( ( finite_card(B,top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_8048_Abs__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A)] :
          ~ ! [Y2: int] :
              ( ( Xd = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y2) )
             => ~ member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ) ).

% Abs_bit1_cases
tff(fact_8049_one__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( one_one(numeral_bit1(A)) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),one_one(int)) ) ) ).

% one_bit1_def
tff(fact_8050_zero__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( zero_zero(numeral_bit1(A)) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),zero_zero(int)) ) ) ).

% zero_bit1_def
tff(fact_8051_bit1_Oof__int__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Z: int] : aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,Z,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.of_int_eq
tff(fact_8052_bit1_Oof__nat__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [K: nat] : aa(nat,numeral_bit1(A),semiring_1_of_nat(numeral_bit1(A)),K) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),K),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.of_nat_eq
tff(fact_8053_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,$o),P,zero_zero(nat)),X_12)
     => ( ! [X: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X)
           => ? [Y3: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N)),Y3)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N),X),Y3) ) )
       => ? [F4: fun(nat,A)] :
          ! [N9: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N9),aa(nat,A,F4,N9))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N9),aa(nat,A,F4,N9)),aa(nat,A,F4,aa(nat,nat,suc,N9))) ) ) ) ).

% dependent_nat_choice
tff(fact_8054_bit1_OUNIV__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( top_top(set(numeral_bit1(A))) = image(int,numeral_bit1(A),numeral_Abs_bit12(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ) ).

% bit1.UNIV_eq
tff(fact_8055_inj__on__Abs__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => inj_on(int,numeral_bit1(A),numeral_Abs_bit12(A),set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% inj_on_Abs_bit1
tff(fact_8056_Abs__bit1__inject,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: int,Y: int] :
          ( member(int,Xd,set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
           => ( ( aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Xd) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y) )
            <=> ( Xd = Y ) ) ) ) ) ).

% Abs_bit1_inject
tff(fact_8057_Abs__bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),Xd: numeral_bit1(A)] :
          ( ! [Y2: int] :
              ( member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
             => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y2)) )
         => aa(numeral_bit1(A),$o,P,Xd) ) ) ).

% Abs_bit1_induct
tff(fact_8058_Abs__bit1__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y)) = Y ) ) ) ).

% Abs_bit1_inverse
tff(fact_8059_Abs__bit1_H__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: int] : numeral_Abs_bit1(A,Xd) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,Xd,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% Abs_bit1'_def
tff(fact_8060_Abs__bit1_H__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: int] : aa(numeral_bit1(A),int,numeral_Rep_bit1(A),numeral_Abs_bit1(A,Xd)) = modulo_modulo(int,Xd,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% Abs_bit1'_code
tff(fact_8061_bit1_ORep__Abs__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),one_one(int))) = one_one(int) ) ) ).

% bit1.Rep_Abs_1
tff(fact_8062_bit1_ORep__Abs__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),zero_zero(int))) = zero_zero(int) ) ) ).

% bit1.Rep_Abs_0
tff(fact_8063_less__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A2: numeral_bit1(A),B2: numeral_bit1(A)] :
          ( aa(numeral_bit1(A),$o,ord_less(numeral_bit1(A),A2),B2)
        <=> aa(int,$o,ord_less(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),A2)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),B2)) ) ) ).

% less_bit1_def
tff(fact_8064_bit1_ORep__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),zero_zero(numeral_bit1(A))) = zero_zero(int) ) ) ).

% bit1.Rep_0
tff(fact_8065_bit1_ORep__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),one_one(numeral_bit1(A))) = one_one(int) ) ) ).

% bit1.Rep_1
tff(fact_8066_bit1_ORep__less__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A)] : aa(int,$o,ord_less(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_less_n
tff(fact_8067_bit1_ORep__mod,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A)] : modulo_modulo(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) = aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd) ) ).

% bit1.Rep_mod
tff(fact_8068_bit1_ORep__numeral,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [W: num] : aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(num,numeral_bit1(A),numeral_numeral(numeral_bit1(A)),W)) = modulo_modulo(int,aa(num,int,numeral_numeral(int),W),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_numeral
tff(fact_8069_bit1_ORep__Abs__mod,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [M: int] : aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,M,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))))) = modulo_modulo(int,M,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_Abs_mod
tff(fact_8070_bit1_Ominus__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),uminus_uminus(numeral_bit1(A)),Xd) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.minus_def
tff(fact_8071_bit1_Oadd__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A),Y: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),aa(numeral_bit1(A),fun(numeral_bit1(A),numeral_bit1(A)),plus_plus(numeral_bit1(A)),Xd),Y) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Y)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.add_def
tff(fact_8072_bit1_Omult__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A),Y: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),aa(numeral_bit1(A),fun(numeral_bit1(A),numeral_bit1(A)),times_times(numeral_bit1(A)),Xd),Y) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Y)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.mult_def
tff(fact_8073_bit1_Odiff__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A),Y: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),minus_minus(numeral_bit1(A),Xd),Y) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),modulo_modulo(int,aa(int,int,minus_minus(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Y)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.diff_def
tff(fact_8074_bit1_OAbs__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [M: int] :
          ( member(int,M,set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))))
         => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),M)) = M ) ) ) ).

% bit1.Abs_inverse
tff(fact_8075_Rep__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit1(A)] : member(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xd),set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% Rep_bit1
tff(fact_8076_Rep__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ~ ! [X: numeral_bit1(A)] : Y != aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X) ) ) ).

% Rep_bit1_cases
tff(fact_8077_Rep__bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int,P: fun(int,$o)] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ( ! [X: numeral_bit1(A)] : aa(int,$o,P,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X))
           => aa(int,$o,P,Y) ) ) ) ).

% Rep_bit1_induct
tff(fact_8078_type__definition__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit1(A),int,numeral_Rep_bit1(A),numeral_Abs_bit12(A),set_or7035219750837199246ssThan(int,zero_zero(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))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% type_definition_bit1
tff(fact_8079_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,ord_less_eq(nat,K),finite_card(A,A3))
       => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_abn(set(A),fun(nat,fun(list(A),$o)),A3),K))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_bg(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,finite_card(A,A3)),K)),one_one(nat)),finite_card(A,A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_8080_distinct__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] :
          ( distinct(A,linorder_sort_key(A,B,F2,Xs))
        <=> distinct(A,Xs) ) ) ).

% distinct_sort
tff(fact_8081_distinct__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))
       => ( distinct(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_8082_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_abn(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_8083_distinct__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs))
    <=> ( distinct(B,Xs)
        & inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xs)) ) ) ).

% distinct_map
tff(fact_8084_not__distinct__decomp,axiom,
    ! [A: $tType,Ws: list(A)] :
      ( ~ distinct(A,Ws)
     => ? [Xs2: list(A),Ys4: list(A),Zs2: list(A),Y2: A] : Ws = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Y2),nil(A))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Y2),nil(A))),Zs2)))) ) ).

% not_distinct_decomp
tff(fact_8085_distinct__singleton,axiom,
    ! [A: $tType,Xd: A] : distinct(A,aa(list(A),list(A),cons(A,Xd),nil(A))) ).

% distinct_singleton
tff(fact_8086_distinct__length__2__or__more,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,A2),aa(list(A),list(A),cons(A,B2),Xs)))
    <=> ( ( A2 != B2 )
        & distinct(A,aa(list(A),list(A),cons(A,A2),Xs))
        & distinct(A,aa(list(A),list(A),cons(A,B2),Xs)) ) ) ).

% distinct_length_2_or_more
tff(fact_8087_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As2: list(A)] :
      ( ~ distinct(A,As2)
    <=> ? [Xs4: list(A),Y5: A,Ys3: list(A)] :
          ( member(A,Y5,aa(list(A),set(A),set2(A),Xs4))
          & distinct(A,Xs4)
          & ( As2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(A),list(A),cons(A,Y5),Ys3)) ) ) ) ).

% not_distinct_conv_prefix
tff(fact_8088_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,Xd: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,Xd),Xs))
    <=> ( ~ member(A,Xd,aa(list(A),set(A),set2(A),Xs))
        & distinct(A,Xs) ) ) ).

% distinct.simps(2)
tff(fact_8089_distinct__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => distinct(A,X) )
     => distinct(list(A),product_lists(A,Xss)) ) ).

% distinct_product_lists
tff(fact_8090_subseqs__distinctD,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => ( distinct(A,Xs)
       => distinct(A,Ys) ) ) ).

% subseqs_distinctD
tff(fact_8091_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J2: nat] :
              ( aa(nat,$o,ord_less(nat,J2),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I2 != J2 )
               => ( aa(nat,A,nth(A,Xs),I2) != aa(nat,A,nth(A,Xs),J2) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_8092_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$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(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_8093_distinct__upt,axiom,
    ! [I: nat,J: nat] : distinct(nat,upt(I,J)) ).

% distinct_upt
tff(fact_8094_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => distinct(A,Xs) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_8095_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( distinct(A,Xs)
     => ( distinct(B,Ys)
       => distinct(product_prod(A,B),product(A,B,Xs,Ys)) ) ) ).

% distinct_product
tff(fact_8096_distinct__upto,axiom,
    ! [I: int,J: int] : distinct(int,upto(I,J)) ).

% distinct_upto
tff(fact_8097_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_8098_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : distinct(A,linord4507533701916653071of_set(A,A3)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_8099_distinct__set__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(set(A),aa(list(list(A)),list(set(A)),map(list(A),set(A),set2(A)),subseqs(A,Xs))) ) ).

% distinct_set_subseqs
tff(fact_8100_distinct__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% distinct_filter
tff(fact_8101_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),P: fun(B,$o)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs))
     => distinct(A,aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xs))) ) ).

% distinct_map_filter
tff(fact_8102_distinct__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),tl(A),Xs)) ) ).

% distinct_tl
tff(fact_8103_finite__distinct__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A3 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_8104_distinct__card,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( finite_card(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% distinct_card
tff(fact_8105_card__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( finite_card(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
     => distinct(A,Xs) ) ).

% card_distinct
tff(fact_8106_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xd: A] :
      ( distinct(A,Xs)
     => ( member(A,Xd,aa(list(A),set(A),set2(A),Xs))
       => ? [X: nat] :
            ( aa(nat,$o,ord_less(nat,X),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X) = Xd )
            & ! [Y3: nat] :
                ( ( aa(nat,$o,ord_less(nat,Y3),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y3) = Xd ) )
               => ( Y3 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_8107_distinct__idx,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),L: list(B),I: nat,J: nat] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),L))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(B),nat,size_size(list(B)),L))
       => ( aa(nat,$o,ord_less(nat,J),aa(list(B),nat,size_size(list(B)),L))
         => ( ( aa(B,A,F2,aa(nat,B,nth(B,L),I)) = aa(B,A,F2,aa(nat,B,nth(B,L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_8108_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X: nat] :
            ( member(nat,X,I6)
           => aa(nat,$o,ord_less(nat,X),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I6) ) ) ).

% inj_on_nth
tff(fact_8109_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B4: set(A)] :
      ( distinct(A,Xs)
     => ( ( A3 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B4 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A3,B4) ) ) ) ).

% bij_betw_nth
tff(fact_8110_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ member(A,A2,aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),insert(A,aa(nat,A,nth(A,Xs),I),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_8111_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xd: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xd)) = insert(A,Xd,aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),insert(A,aa(nat,A,nth(A,Xs),Nb),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_8112_bit1_Otype,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit1(A),int,numeral_Rep_bit1(A),numeral_Abs_bit12(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.type
tff(fact_8113_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( aa(nat,$o,ord_less(nat,K),finite_card(A,A3))
     => ( finite_card(list(A),collect(list(A),aa(set(A),fun(list(A),$o),aTP_Lamp_abo(nat,fun(set(A),fun(list(A),$o)),K),A3))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_bg(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,finite_card(A,A3)),K)),one_one(nat)),finite_card(A,A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_8114_mod__type__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
        <=> ( type_definition(A,int,Rep,Abs,set_or7035219750837199246ssThan(int,zero_zero(int),Nb))
            & aa(int,$o,ord_less(int,one_one(int)),Nb)
            & ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) )
            & ( one_one(A) = aa(int,A,Abs,one_one(int)) )
            & ! [X5: A,Y5: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Y5) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X5)),aa(A,int,Rep,Y5)),Nb))
            & ! [X5: A,Y5: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X5),Y5) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X5)),aa(A,int,Rep,Y5)),Nb))
            & ! [X5: A,Y5: A] : aa(A,A,minus_minus(A,X5),Y5) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,minus_minus(int,aa(A,int,Rep,X5)),aa(A,int,Rep,Y5)),Nb))
            & ! [X5: A] : aa(A,A,uminus_uminus(A),X5) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(A,int,Rep,X5)),Nb)) ) ) ) ).

% mod_type_def
tff(fact_8115_mod__type_Ointro,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Rep: fun(A,int),Abs: fun(int,A),Nb: int] :
          ( type_definition(A,int,Rep,Abs,set_or7035219750837199246ssThan(int,zero_zero(int),Nb))
         => ( aa(int,$o,ord_less(int,one_one(int)),Nb)
           => ( ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) )
             => ( ( one_one(A) = aa(int,A,Abs,one_one(int)) )
               => ( ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y2) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X)),aa(A,int,Rep,Y2)),Nb))
                 => ( ! [X: A,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),Y2) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X)),aa(A,int,Rep,Y2)),Nb))
                   => ( ! [X: A,Y2: A] : aa(A,A,minus_minus(A,X),Y2) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,minus_minus(int,aa(A,int,Rep,X)),aa(A,int,Rep,Y2)),Nb))
                     => ( ! [X: A] : aa(A,A,uminus_uminus(A),X) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(A,int,Rep,X)),Nb))
                       => numeral_mod_type(A,Nb,Rep,Abs) ) ) ) ) ) ) ) ) ) ).

% mod_type.intro
tff(fact_8116_mod__type_Osize0,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => aa(int,$o,ord_less(int,zero_zero(int)),Nb) ) ) ).

% mod_type.size0
tff(fact_8117_mod__type_Osize1,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => aa(int,$o,ord_less(int,one_one(int)),Nb) ) ) ).

% mod_type.size1
tff(fact_8118_mod__type_ORep__less__n,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),Xd: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => aa(int,$o,ord_less(int,aa(A,int,Rep,Xd)),Nb) ) ) ).

% mod_type.Rep_less_n
tff(fact_8119_mod__type_ORep__mod,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),Xd: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( modulo_modulo(int,aa(A,int,Rep,Xd),Nb) = aa(A,int,Rep,Xd) ) ) ) ).

% mod_type.Rep_mod
tff(fact_8120_mod__type_ORep__Abs__mod,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),M: int] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,int,Rep,aa(int,A,Abs,modulo_modulo(int,M,Nb))) = modulo_modulo(int,M,Nb) ) ) ) ).

% mod_type.Rep_Abs_mod
tff(fact_8121_mod__type_OAbs__inverse,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),M: int] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( member(int,M,set_or7035219750837199246ssThan(int,zero_zero(int),Nb))
           => ( aa(A,int,Rep,aa(int,A,Abs,M)) = M ) ) ) ) ).

% mod_type.Abs_inverse
tff(fact_8122_mod__type_ORep__1,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,int,Rep,one_one(A)) = one_one(int) ) ) ) ).

% mod_type.Rep_1
tff(fact_8123_mod__type_Oone__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( one_one(A) = aa(int,A,Abs,one_one(int)) ) ) ) ).

% mod_type.one_def
tff(fact_8124_mod__type_ORep__0,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,int,Rep,zero_zero(A)) = zero_zero(int) ) ) ) ).

% mod_type.Rep_0
tff(fact_8125_mod__type_Ozero__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) ) ) ) ).

% mod_type.zero_def
tff(fact_8126_mod__type_ORep__Abs__1,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,int,Rep,aa(int,A,Abs,one_one(int))) = one_one(int) ) ) ) ).

% mod_type.Rep_Abs_1
tff(fact_8127_mod__type_ORep__Abs__0,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,int,Rep,aa(int,A,Abs,zero_zero(int))) = zero_zero(int) ) ) ) ).

% mod_type.Rep_Abs_0
tff(fact_8128_mod__type_Oadd__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),Xd: A,Y: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xd),Y) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,Xd)),aa(A,int,Rep,Y)),Nb)) ) ) ) ).

% mod_type.add_def
tff(fact_8129_mod__type_Omult__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),Xd: A,Y: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),Xd),Y) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,Xd)),aa(A,int,Rep,Y)),Nb)) ) ) ) ).

% mod_type.mult_def
tff(fact_8130_mod__type_Ominus__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),Xd: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,A,uminus_uminus(A),Xd) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(A,int,Rep,Xd)),Nb)) ) ) ) ).

% mod_type.minus_def
tff(fact_8131_mod__type_Odiff__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),Xd: A,Y: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,A,minus_minus(A,Xd),Y) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,minus_minus(int,aa(A,int,Rep,Xd)),aa(A,int,Rep,Y)),Nb)) ) ) ) ).

% mod_type.diff_def
tff(fact_8132_mod__type_Otype,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => type_definition(A,int,Rep,Abs,set_or7035219750837199246ssThan(int,zero_zero(int),Nb)) ) ) ).

% mod_type.type
tff(fact_8133_bit1__iszero__numeral,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [W: num] :
          ( ring_1_iszero(numeral_bit1(A),aa(num,numeral_bit1(A),numeral_numeral(numeral_bit1(A)),W))
        <=> ( modulo_modulo(int,aa(num,int,numeral_numeral(int),W),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) = zero_zero(int) ) ) ) ).

% bit1_iszero_numeral
tff(fact_8134_Abs__bit0__inject,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: int,Y: int] :
          ( member(int,Xd,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))
         => ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))
           => ( ( aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Xd) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y) )
            <=> ( Xd = Y ) ) ) ) ) ).

% Abs_bit0_inject
tff(fact_8135_iszero__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: num] :
          ( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ) ).

% iszero_neg_numeral
tff(fact_8136_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_8137_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_8138_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_8139_zero__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( zero_zero(numeral_bit0(A)) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),zero_zero(int)) ) ) ).

% zero_bit0_def
tff(fact_8140_one__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( one_one(numeral_bit0(A)) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),one_one(int)) ) ) ).

% one_bit0_def
tff(fact_8141_not__iszero__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),one2)) ) ).

% not_iszero_Numeral1
tff(fact_8142_not__iszero__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ).

% not_iszero_numeral
tff(fact_8143_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: A,Y: A] :
          ( ( Xd = Y )
        <=> ring_1_iszero(A,aa(A,A,minus_minus(A,Xd),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_8144_not__iszero__neg__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_iszero_neg_1
tff(fact_8145_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_8146_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: num] :
          ( ( aa(num,A,numeral_numeral(A),Xd) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xd)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_8147_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_8148_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xd)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xd)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_8149_not__iszero__neg__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) ) ).

% not_iszero_neg_Numeral1
tff(fact_8150_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),Xd) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_8151_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xd)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_8152_bit0_Oof__nat__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [K: nat] : aa(nat,numeral_bit0(A),semiring_1_of_nat(numeral_bit0(A)),K) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),K),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.of_nat_eq
tff(fact_8153_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_8154_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xd: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xd)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_8155_bit0_OUNIV__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( top_top(set(numeral_bit0(A))) = image(int,numeral_bit0(A),numeral_Abs_bit02(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ) ).

% bit0.UNIV_eq
tff(fact_8156_bit0_Oof__int__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Z: int] : aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),modulo_modulo(int,Z,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.of_int_eq
tff(fact_8157_inj__on__Abs__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => inj_on(int,numeral_bit0(A),numeral_Abs_bit02(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))) ) ).

% inj_on_Abs_bit0
tff(fact_8158_Abs__bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: numeral_bit0(A)] :
          ~ ! [Y2: int] :
              ( ( Xd = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y2) )
             => ~ member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))) ) ) ).

% Abs_bit0_cases
tff(fact_8159_Abs__bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit0(A),$o),Xd: numeral_bit0(A)] :
          ( ! [Y2: int] :
              ( member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))
             => aa(numeral_bit0(A),$o,P,aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y2)) )
         => aa(numeral_bit0(A),$o,P,Xd) ) ) ).

% Abs_bit0_induct
tff(fact_8160_Abs__bit0_H__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: int] : numeral_Abs_bit0(A,Xd) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),modulo_modulo(int,Xd,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% Abs_bit0'_def
tff(fact_8161_Abs__bit0__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))
         => ( numeral_Rep_bit0(A,aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y)) = Y ) ) ) ).

% Abs_bit0_inverse
tff(fact_8162_word__eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( zero_zero(word(A)) = aa(num,word(A),numeral_numeral(word(A)),Y) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Y)) ) ) ).

% word_eq_numeral_iff_iszero(10)
tff(fact_8163_word__eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),Xd) = zero_zero(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Xd)) ) ) ).

% word_eq_numeral_iff_iszero(9)
tff(fact_8164_word__eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( zero_zero(word(A)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Y)) ) ) ).

% word_eq_numeral_iff_iszero(12)
tff(fact_8165_word__eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] :
          ( ( aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xd)) = zero_zero(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Xd)) ) ) ).

% word_eq_numeral_iff_iszero(11)
tff(fact_8166_word__eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num,Y: num] :
          ( ( aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xd)) = aa(num,word(A),numeral_numeral(word(A)),Y) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),Y))) ) ) ).

% word_eq_numeral_iff_iszero(3)
tff(fact_8167_word__eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num,Y: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),Xd) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),Y))) ) ) ).

% word_eq_numeral_iff_iszero(2)
tff(fact_8168_iszero__word__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Bin))
        <=> ring_1_iszero(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin))) ) ) ).

% iszero_word_no
tff(fact_8169_word__eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( one_one(word(A)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% word_eq_numeral_iff_iszero(8)
tff(fact_8170_word__eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xd: num] :
          ( ( aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xd)) = one_one(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xd),one2))) ) ) ).

% word_eq_numeral_iff_iszero(7)
tff(fact_8171_bit0__iszero__numeral,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [W: num] :
          ( ring_1_iszero(numeral_bit0(A),aa(num,numeral_bit0(A),numeral_numeral(numeral_bit0(A)),W))
        <=> ( modulo_modulo(int,aa(num,int,numeral_numeral(int),W),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) = zero_zero(int) ) ) ) ).

% bit0_iszero_numeral
tff(fact_8172_bit0_ORep__Abs__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( numeral_Rep_bit0(A,aa(int,numeral_bit0(A),numeral_Abs_bit02(A),zero_zero(int))) = zero_zero(int) ) ) ).

% bit0.Rep_Abs_0
tff(fact_8173_bit0_ORep__Abs__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( numeral_Rep_bit0(A,aa(int,numeral_bit0(A),numeral_Abs_bit02(A),one_one(int))) = one_one(int) ) ) ).

% bit0.Rep_Abs_1
tff(fact_8174_less__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A2: numeral_bit0(A),B2: numeral_bit0(A)] :
          ( aa(numeral_bit0(A),$o,ord_less(numeral_bit0(A),A2),B2)
        <=> aa(int,$o,ord_less(int,numeral_Rep_bit0(A,A2)),numeral_Rep_bit0(A,B2)) ) ) ).

% less_bit0_def
tff(fact_8175_bit0_ORep__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( numeral_Rep_bit0(A,one_one(numeral_bit0(A))) = one_one(int) ) ) ).

% bit0.Rep_1
tff(fact_8176_bit0_ORep__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( numeral_Rep_bit0(A,zero_zero(numeral_bit0(A))) = zero_zero(int) ) ) ).

% bit0.Rep_0
tff(fact_8177_Abs__bit0_H__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xd: int] : numeral_Rep_bit0(A,numeral_Abs_bit0(A,Xd)) = modulo_modulo(int,Xd,aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% Abs_bit0'_code
tff(fact_8178_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_ci(nat,real),Uu) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8179_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_sh(A,A),Uu) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,exp(A),Uu)),one_one(A)),Uu) ) ).

% ATP.lambda_2
tff(fact_8180_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_dz(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_3
tff(fact_8181_ATP_Olambda__4,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_ko(real,$o),Uu)
    <=> ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Uu)
        & aa(real,$o,ord_less_eq(real,Uu),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_4
tff(fact_8182_ATP_Olambda__5,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tq(nat,A),Uu) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu)),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_5
tff(fact_8183_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_fv(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_6
tff(fact_8184_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tr(nat,A),Uu) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_7
tff(fact_8185_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_sl(real,real),Uu) = divide_divide(real,cos(real,Uu),sin(real,Uu)) ).

% ATP.lambda_8
tff(fact_8186_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_vp(real,real),Uu) = divide_divide(real,aa(real,real,ln_ln(real),Uu),Uu) ).

% ATP.lambda_9
tff(fact_8187_ATP_Olambda__10,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_mg(word(B),$o),Uu)
        <=> aa(word(B),$o,ord_less(word(B),Uu),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% ATP.lambda_10
tff(fact_8188_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_tc(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_11
tff(fact_8189_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_lj(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_12
tff(fact_8190_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_jh(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_13
tff(fact_8191_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_zk(A,list(A)),Uu) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_14
tff(fact_8192_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_tk(nat,real),Uu) = divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_15
tff(fact_8193_ATP_Olambda__16,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tp(nat,A),Uu) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_16
tff(fact_8194_ATP_Olambda__17,axiom,
    ! [Uu: int] :
      ( aa(int,$o,aTP_Lamp_ku(int,$o),Uu)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Uu) ) ).

% ATP.lambda_17
tff(fact_8195_ATP_Olambda__18,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_aaf(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_18
tff(fact_8196_ATP_Olambda__19,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_aag(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_19
tff(fact_8197_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aaj(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_20
tff(fact_8198_ATP_Olambda__21,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_oa(real,real),Uu) = suminf(real,aTP_Lamp_cg(real,fun(nat,real),Uu)) ).

% ATP.lambda_21
tff(fact_8199_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_tl(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_22
tff(fact_8200_ATP_Olambda__23,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tf(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_23
tff(fact_8201_ATP_Olambda__24,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_aae(list(B),fun(nat,nat)),Uu) = ord_max(nat,aa(list(B),nat,size_size(list(B)),Uu)) ).

% ATP.lambda_24
tff(fact_8202_ATP_Olambda__25,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_zj(list(A),fun(nat,nat)),Uu) = ord_max(nat,aa(list(A),nat,size_size(list(A)),Uu)) ).

% ATP.lambda_25
tff(fact_8203_ATP_Olambda__26,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_yt(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_26
tff(fact_8204_ATP_Olambda__27,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_ys(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).

% ATP.lambda_27
tff(fact_8205_ATP_Olambda__28,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_kc(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ).

% ATP.lambda_28
tff(fact_8206_ATP_Olambda__29,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_xo(product_prod(A,A),$o),Uu)
        <=> ? [X5: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X5),Y5) )
              & aa(A,$o,ord_less(A,X5),Y5) ) ) ) ).

% ATP.lambda_29
tff(fact_8207_ATP_Olambda__30,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_xp(product_prod(A,A),$o),Uu)
        <=> ? [X5: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X5),Y5) )
              & aa(A,$o,ord_less(A,Y5),X5) ) ) ) ).

% ATP.lambda_30
tff(fact_8208_ATP_Olambda__31,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_yx(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_31
tff(fact_8209_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_jp(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_32
tff(fact_8210_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gx(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_33
tff(fact_8211_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_jo(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_34
tff(fact_8212_ATP_Olambda__35,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_ea(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,divide_divide(nat,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_35
tff(fact_8213_ATP_Olambda__36,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] :
      aa(nat,option(A),aTP_Lamp_kw(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_36
tff(fact_8214_ATP_Olambda__37,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_xh(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,Uu)),Uua)),abs_abs(int,Uu))) ).

% ATP.lambda_37
tff(fact_8215_ATP_Olambda__38,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gw(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_8216_ATP_Olambda__39,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_yo(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$true,aTP_Lamp_yn(A,fun(A,$o),Uua),Uu) ) ) ).

% ATP.lambda_39
tff(fact_8217_ATP_Olambda__40,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_yv(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ys(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_40
tff(fact_8218_ATP_Olambda__41,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_zi(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zh(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_41
tff(fact_8219_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kk(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_42
tff(fact_8220_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ki(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_kh(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_43
tff(fact_8221_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_cb(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)),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_44
tff(fact_8222_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ds(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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_45
tff(fact_8223_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ec(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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_46
tff(fact_8224_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ia(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_47
tff(fact_8225_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_jd(nat,fun(nat,A),Uu),Uua) = 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_48
tff(fact_8226_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_kp(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_49
tff(fact_8227_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ke(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,ord_less(real,Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_50
tff(fact_8228_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_kq(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_51
tff(fact_8229_ATP_Olambda__52,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_kd(complex,fun(real,$o),Uu),Uua)
    <=> ( ( sgn_sgn(complex,Uu) = cis(Uua) )
        & aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),pi) ) ) ).

% ATP.lambda_52
tff(fact_8230_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_cg(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),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_53
tff(fact_8231_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ob(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_54
tff(fact_8232_ATP_Olambda__55,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_55
tff(fact_8233_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ie(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_56
tff(fact_8234_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_iv(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_57
tff(fact_8235_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_it(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,divide_divide(A,Uu,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_58
tff(fact_8236_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_iw(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_59
tff(fact_8237_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hu(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_60
tff(fact_8238_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gs(real,fun(nat,real),Uu),Uua) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_61
tff(fact_8239_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_vr(nat,fun(real,real),Uu),Uua) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_62
tff(fact_8240_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hd(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_63
tff(fact_8241_ATP_Olambda__64,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hc(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_64
tff(fact_8242_ATP_Olambda__65,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_gb(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_65
tff(fact_8243_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_al(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_66
tff(fact_8244_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_tx(real,fun(nat,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_67
tff(fact_8245_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_vb(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Uu),Uua)),divide_divide(real,one_one(real),Uua)) ).

% ATP.lambda_68
tff(fact_8246_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_vs(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,Uu,Uua)),Uua) ).

% ATP.lambda_69
tff(fact_8247_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_av(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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_70
tff(fact_8248_ATP_Olambda__71,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_kn(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),pi)
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_71
tff(fact_8249_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bk(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_72
tff(fact_8250_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fn(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_73
tff(fact_8251_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tg(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_74
tff(fact_8252_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ey(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_75
tff(fact_8253_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ck(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_76
tff(fact_8254_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fa(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_77
tff(fact_8255_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dd(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_78
tff(fact_8256_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ce(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_79
tff(fact_8257_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ek(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_80
tff(fact_8258_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dc(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_81
tff(fact_8259_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fl(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_82
tff(fact_8260_ATP_Olambda__83,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fk(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_83
tff(fact_8261_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_th(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_84
tff(fact_8262_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gd(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_85
tff(fact_8263_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hb(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_86
tff(fact_8264_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ha(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_87
tff(fact_8265_ATP_Olambda__88,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wl(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less(real,aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_88
tff(fact_8266_ATP_Olambda__89,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_qa(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_89
tff(fact_8267_ATP_Olambda__90,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_jq(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_90
tff(fact_8268_ATP_Olambda__91,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_zf(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),cons(A,aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_91
tff(fact_8269_ATP_Olambda__92,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ev(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_92
tff(fact_8270_ATP_Olambda__93,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bz(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_93
tff(fact_8271_ATP_Olambda__94,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_uh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_94
tff(fact_8272_ATP_Olambda__95,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ug(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_sr(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_95
tff(fact_8273_ATP_Olambda__96,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_lr(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_lq(list(A),fun(A,list(A)),Uua)),Uu) ).

% ATP.lambda_96
tff(fact_8274_ATP_Olambda__97,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_up(fun(A,B),fun(A,real),Uu),Uua) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_97
tff(fact_8275_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jg(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_98
tff(fact_8276_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jm(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_99
tff(fact_8277_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jj(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_100
tff(fact_8278_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ji(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_101
tff(fact_8279_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ed(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_102
tff(fact_8280_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: num,Uua: num] : aa(num,word(A),aTP_Lamp_yq(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_103
tff(fact_8281_ATP_Olambda__104,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_yr(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_104
tff(fact_8282_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_is(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_105
tff(fact_8283_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_mk(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,Uua)),Uu)) ).

% ATP.lambda_106
tff(fact_8284_ATP_Olambda__107,axiom,
    ! [Uu: $o,Uua: nat] : aa(nat,nat,aa($o,fun(nat,nat),aTP_Lamp_zg($o,fun(nat,nat)),(Uu)),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa($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_107
tff(fact_8285_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_in(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_108
tff(fact_8286_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_io(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_109
tff(fact_8287_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_uc(A,fun(nat,A),Uu),Uua) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_110
tff(fact_8288_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ub(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_111
tff(fact_8289_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gv(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_112
tff(fact_8290_ATP_Olambda__113,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fu(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_113
tff(fact_8291_ATP_Olambda__114,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_ln(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),cons(A,Uua)),product_lists(A,Uu)) ).

% ATP.lambda_114
tff(fact_8292_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ja(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_115
tff(fact_8293_ATP_Olambda__116,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_lo(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),product_Pair(A,B,Uua)),Uu) ).

% ATP.lambda_116
tff(fact_8294_ATP_Olambda__117,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_jt(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(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_117
tff(fact_8295_ATP_Olambda__118,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_jv(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(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_118
tff(fact_8296_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bb(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_119
tff(fact_8297_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_zx(list(A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),aa(nat,A,nth(A,Uu),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% ATP.lambda_120
tff(fact_8298_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_zy(list(A),fun(A,$o),Uu),Uua)
        <=> ( Uua = aa(nat,A,nth(A,Uu),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% ATP.lambda_121
tff(fact_8299_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_id(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_122
tff(fact_8300_ATP_Olambda__123,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_tz(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))))) ).

% ATP.lambda_123
tff(fact_8301_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_124
tff(fact_8302_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lb(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_125
tff(fact_8303_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_tw(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_126
tff(fact_8304_ATP_Olambda__127,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_tv(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_127
tff(fact_8305_ATP_Olambda__128,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_tn(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_128
tff(fact_8306_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_td(A,fun(nat,A),Uu),Uua) = divide_divide(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_129
tff(fact_8307_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aai(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_130
tff(fact_8308_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_iz(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_131
tff(fact_8309_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_iy(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_132
tff(fact_8310_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bi(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_133
tff(fact_8311_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aax(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less_eq(nat,Uua),Uu) ) ).

% ATP.lambda_134
tff(fact_8312_ATP_Olambda__135,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mi(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_135
tff(fact_8313_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_uk(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_136
tff(fact_8314_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lv(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_137
tff(fact_8315_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mn(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_138
tff(fact_8316_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_as(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_8317_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_vy(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_140
tff(fact_8318_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_yn(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_141
tff(fact_8319_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_fw(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_142
tff(fact_8320_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_tb(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_143
tff(fact_8321_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_uj(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_8322_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ah(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_8323_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lw(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_8324_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_md(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_147
tff(fact_8325_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_nl(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_148
tff(fact_8326_ATP_Olambda__149,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_mc(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_149
tff(fact_8327_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_li(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_150
tff(fact_8328_ATP_Olambda__151,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_mb(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_151
tff(fact_8329_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mo(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_152
tff(fact_8330_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_8331_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aTP_Lamp_lf(list(A),fun(array(A),assn),Uu),Uua) = aa(list(A),assn,snga_assn(A,Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_8332_ATP_Olambda__155,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_no(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_155
tff(fact_8333_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ak(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,dvd_dvd(nat,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_8334_ATP_Olambda__157,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_ai(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,dvd_dvd(int,Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_8335_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ae(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,dvd_dvd(A,Uua),Uu) ) ) ).

% ATP.lambda_158
tff(fact_8336_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gy(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_159
tff(fact_8337_ATP_Olambda__160,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aTP_Lamp_lq(list(A),fun(A,list(A)),Uu),Uua) = aa(list(A),list(A),cons(A,Uua),Uu) ).

% ATP.lambda_160
tff(fact_8338_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_tm(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_161
tff(fact_8339_ATP_Olambda__162,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_a(set(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_162
tff(fact_8340_ATP_Olambda__163,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_aah(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_163
tff(fact_8341_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_zn(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_164
tff(fact_8342_ATP_Olambda__165,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_zo(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),cons(list(A),Uua),nil(list(A))) ).

% ATP.lambda_165
tff(fact_8343_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ww(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_166
tff(fact_8344_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_wn(fun(A,real),fun(A,$o),Uu),Uua)
        <=> aa(real,$o,ord_less_eq(real,one_one(real)),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_167
tff(fact_8345_ATP_Olambda__168,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wr(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less_eq(real,one_one(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_168
tff(fact_8346_ATP_Olambda__169,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wk(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_169
tff(fact_8347_ATP_Olambda__170,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ay(fun(A,nat),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_170
tff(fact_8348_ATP_Olambda__171,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_gr(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_171
tff(fact_8349_ATP_Olambda__172,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_gq(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_172
tff(fact_8350_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_vg(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_173
tff(fact_8351_ATP_Olambda__174,axiom,
    ! [Uu: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_wq(fun(real,$o),fun(real,$o),Uu),Uua)
    <=> aa(real,$o,Uu,aa(real,real,inverse_inverse(real),Uua)) ) ).

% ATP.lambda_174
tff(fact_8352_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_vq(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_175
tff(fact_8353_ATP_Olambda__176,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ml(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_176
tff(fact_8354_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_177
tff(fact_8355_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_sy(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_178
tff(fact_8356_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_179
tff(fact_8357_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_180
tff(fact_8358_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_et(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_181
tff(fact_8359_ATP_Olambda__182,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ll(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_182
tff(fact_8360_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_yp(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_183
tff(fact_8361_ATP_Olambda__184,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_yu(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_184
tff(fact_8362_ATP_Olambda__185,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_oe(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_od(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_185
tff(fact_8363_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_nm(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_186
tff(fact_8364_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_js(nat,fun(nat,complex),Uu),Uua) = cis(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_187
tff(fact_8365_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_oy(A,fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu)),Uua)),aa(A,A,inverse_inverse(A),Uu))) ) ).

% ATP.lambda_188
tff(fact_8366_ATP_Olambda__189,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_vu(fun(real,real),fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_189
tff(fact_8367_ATP_Olambda__190,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aak(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_190
tff(fact_8368_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_jk(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_191
tff(fact_8369_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ao(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),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_192
tff(fact_8370_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ir(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_193
tff(fact_8371_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_ip(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_194
tff(fact_8372_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_iq(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_195
tff(fact_8373_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_jf(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_196
tff(fact_8374_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ap(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_197
tff(fact_8375_ATP_Olambda__198,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_tt(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_198
tff(fact_8376_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mv(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_199
tff(fact_8377_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zb(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uu,Uua)) ).

% ATP.lambda_200
tff(fact_8378_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zc(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uua,Uu)) ).

% ATP.lambda_201
tff(fact_8379_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yy(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uu),Uua)) ).

% ATP.lambda_202
tff(fact_8380_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yz(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uua),Uu)) ).

% ATP.lambda_203
tff(fact_8381_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_zw(A,fun(A,$o),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_204
tff(fact_8382_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_hj(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_205
tff(fact_8383_ATP_Olambda__206,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_rz(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_206
tff(fact_8384_ATP_Olambda__207,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_tj(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_207
tff(fact_8385_ATP_Olambda__208,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ow(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_208
tff(fact_8386_ATP_Olambda__209,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xk(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_209
tff(fact_8387_ATP_Olambda__210,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_210
tff(fact_8388_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_my(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_211
tff(fact_8389_ATP_Olambda__212,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_se(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_212
tff(fact_8390_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_213
tff(fact_8391_ATP_Olambda__214,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_rp(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_214
tff(fact_8392_ATP_Olambda__215,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_cd(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_215
tff(fact_8393_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_216
tff(fact_8394_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ya(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_217
tff(fact_8395_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ta(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_218
tff(fact_8396_ATP_Olambda__219,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_gc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_219
tff(fact_8397_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_bh(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_220
tff(fact_8398_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ff(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_221
tff(fact_8399_ATP_Olambda__222,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_yj(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,artanh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_222
tff(fact_8400_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_223
tff(fact_8401_ATP_Olambda__224,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_224
tff(fact_8402_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_225
tff(fact_8403_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_of(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_226
tff(fact_8404_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_227
tff(fact_8405_ATP_Olambda__228,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_ye(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_228
tff(fact_8406_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_229
tff(fact_8407_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ry(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_230
tff(fact_8408_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_px(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_231
tff(fact_8409_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_232
tff(fact_8410_ATP_Olambda__233,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xn(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_233
tff(fact_8411_ATP_Olambda__234,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sf(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_234
tff(fact_8412_ATP_Olambda__235,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_rm(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_235
tff(fact_8413_ATP_Olambda__236,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ej(fun(B,A),fun(B,A),Uu),Uua) = abs_abs(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_8414_ATP_Olambda__237,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rs(fun(A,real),fun(A,real),Uu),Uua) = abs_abs(real,aa(A,real,Uu,Uua)) ).

% ATP.lambda_237
tff(fact_8415_ATP_Olambda__238,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yd(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_238
tff(fact_8416_ATP_Olambda__239,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sj(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_239
tff(fact_8417_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ns(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_240
tff(fact_8418_ATP_Olambda__241,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_rq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_241
tff(fact_8419_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_nd(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_242
tff(fact_8420_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_nc(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_243
tff(fact_8421_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pt(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_244
tff(fact_8422_ATP_Olambda__245,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rr(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_245
tff(fact_8423_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_os(fun(A,real),fun(A,real),Uu),Uua) = sin(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_246
tff(fact_8424_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_mt(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_8425_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_oq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_8426_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_mq(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_8427_ATP_Olambda__250,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sa(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_8428_ATP_Olambda__251,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_uf(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_251
tff(fact_8429_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pb(fun(A,real),fun(A,real),Uu),Uua) = cos(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_8430_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_mw(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_8431_ATP_Olambda__254,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_pw(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_254
tff(fact_8432_ATP_Olambda__255,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_sk(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_255
tff(fact_8433_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_256
tff(fact_8434_ATP_Olambda__257,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xy(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_257
tff(fact_8435_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_258
tff(fact_8436_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_259
tff(fact_8437_ATP_Olambda__260,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_abd(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_260
tff(fact_8438_ATP_Olambda__261,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_kf(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_261
tff(fact_8439_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_sn(fun(real,A),fun(real,real),Uu),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu,Uua))) ) ).

% ATP.lambda_262
tff(fact_8440_ATP_Olambda__263,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_yi(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_263
tff(fact_8441_ATP_Olambda__264,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_xl(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [M4: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Uua),M4)
         => ( aa(nat,$o,Uu,M4)
          <=> aa(nat,$o,Uu,Uua) ) ) ) ).

% ATP.lambda_264
tff(fact_8442_ATP_Olambda__265,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ze(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [N10: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Uua),N10)
         => aa(nat,$o,Uu,N10) ) ) ).

% ATP.lambda_265
tff(fact_8443_ATP_Olambda__266,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_zd(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [N10: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Uua),N10)
         => ~ aa(nat,$o,Uu,N10) ) ) ).

% ATP.lambda_266
tff(fact_8444_ATP_Olambda__267,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_eb(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,divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu,divide_divide(nat,aa(nat,nat,minus_minus(nat,Uub),one_one(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_267
tff(fact_8445_ATP_Olambda__268,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gp(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_268
tff(fact_8446_ATP_Olambda__269,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_kb(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),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),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_269
tff(fact_8447_ATP_Olambda__270,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_jy(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),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),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_270
tff(fact_8448_ATP_Olambda__271,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jz(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),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),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_271
tff(fact_8449_ATP_Olambda__272,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_ka(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),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),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_272
tff(fact_8450_ATP_Olambda__273,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_ct(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_273
tff(fact_8451_ATP_Olambda__274,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_eh(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_274
tff(fact_8452_ATP_Olambda__275,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_bn(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_275
tff(fact_8453_ATP_Olambda__276,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_co(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_276
tff(fact_8454_ATP_Olambda__277,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_ei(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_277
tff(fact_8455_ATP_Olambda__278,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_bm(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_278
tff(fact_8456_ATP_Olambda__279,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_cu(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_279
tff(fact_8457_ATP_Olambda__280,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,$o),fun(A,B),aTP_Lamp_es(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_280
tff(fact_8458_ATP_Olambda__281,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,$o),fun(A,B),aTP_Lamp_by(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_281
tff(fact_8459_ATP_Olambda__282,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_aao(fun(B,A),fun(fun(B,$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_282
tff(fact_8460_ATP_Olambda__283,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( aa(real,$o,aa(fun(real,real),fun(real,$o),aTP_Lamp_vv(fun(real,real),fun(fun(real,real),fun(real,$o)),Uu),Uua),Uub)
    <=> has_field_derivative(real,Uu,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_283
tff(fact_8461_ATP_Olambda__284,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_kj(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_284
tff(fact_8462_ATP_Olambda__285,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_kh(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_285
tff(fact_8463_ATP_Olambda__286,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_pv(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_286
tff(fact_8464_ATP_Olambda__287,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_cl(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_287
tff(fact_8465_ATP_Olambda__288,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_fb(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_288
tff(fact_8466_ATP_Olambda__289,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ri(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_289
tff(fact_8467_ATP_Olambda__290,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_rk(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_290
tff(fact_8468_ATP_Olambda__291,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_pj(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_291
tff(fact_8469_ATP_Olambda__292,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_im(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_il(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_292
tff(fact_8470_ATP_Olambda__293,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_ik(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_ij(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_293
tff(fact_8471_ATP_Olambda__294,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_ii(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_ih(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_294
tff(fact_8472_ATP_Olambda__295,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_hl(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_hk(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_295
tff(fact_8473_ATP_Olambda__296,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_dy(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_296
tff(fact_8474_ATP_Olambda__297,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_cm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cl(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_297
tff(fact_8475_ATP_Olambda__298,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_fc(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_fb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_298
tff(fact_8476_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_sm(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_da(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_299
tff(fact_8477_ATP_Olambda__300,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_pk(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_pj(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_300
tff(fact_8478_ATP_Olambda__301,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_nt(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uub),zero_zero(real)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_301
tff(fact_8479_ATP_Olambda__302,axiom,
    ! [Uu: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_nu(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_302
tff(fact_8480_ATP_Olambda__303,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_gn(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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_303
tff(fact_8481_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_hh(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_304
tff(fact_8482_ATP_Olambda__305,axiom,
    ! [Uu: array(vEBT_VEBTi),Uua: list(vEBT_VEBTi),Uub: vEBT_VEBTi] : aa(vEBT_VEBTi,assn,aa(list(vEBT_VEBTi),fun(vEBT_VEBTi,assn),aTP_Lamp_dk(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),fun(vEBT_VEBTi,assn)),Uu),Uua),Uub) = 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,Uu),Uua)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,replicate(vEBT_VEBT,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))),divide_divide(nat,na,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,na,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),Uua))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Uub)) ).

% ATP.lambda_305
tff(fact_8483_ATP_Olambda__306,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_sg(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua)),Uub) ) ).

% ATP.lambda_306
tff(fact_8484_ATP_Olambda__307,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_sb(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua)),Uub) ) ).

% ATP.lambda_307
tff(fact_8485_ATP_Olambda__308,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_el(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = 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_308
tff(fact_8486_ATP_Olambda__309,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_ql(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua)),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_309
tff(fact_8487_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_sc(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua)),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_310
tff(fact_8488_ATP_Olambda__311,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_oc(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_311
tff(fact_8489_ATP_Olambda__312,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_go(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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_312
tff(fact_8490_ATP_Olambda__313,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_ef(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_313
tff(fact_8491_ATP_Olambda__314,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_abo(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_314
tff(fact_8492_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_abn(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_315
tff(fact_8493_ATP_Olambda__316,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_lp(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_316
tff(fact_8494_ATP_Olambda__317,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_an(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_317
tff(fact_8495_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_am(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(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_318
tff(fact_8496_ATP_Olambda__319,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_np(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_319
tff(fact_8497_ATP_Olambda__320,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aba(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu)
        & aa(nat,$o,ord_less(nat,Uub),Uua) ) ) ).

% ATP.lambda_320
tff(fact_8498_ATP_Olambda__321,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_abi(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,ord_less(nat,Uub),aa(list(A),nat,size_size(list(A)),Uua))
        & aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_321
tff(fact_8499_ATP_Olambda__322,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_zu(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_322
tff(fact_8500_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ix(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = 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_323
tff(fact_8501_ATP_Olambda__324,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_kv(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_324
tff(fact_8502_ATP_Olambda__325,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_abb(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,ord_less(nat,Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_325
tff(fact_8503_ATP_Olambda__326,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_ee(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_326
tff(fact_8504_ATP_Olambda__327,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_fh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_327
tff(fact_8505_ATP_Olambda__328,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_fo(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_328
tff(fact_8506_ATP_Olambda__329,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hf(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_329
tff(fact_8507_ATP_Olambda__330,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ad(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uub)
        & aa(int,$o,ord_less(int,Uub),Uua) ) ) ).

% ATP.lambda_330
tff(fact_8508_ATP_Olambda__331,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ac(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less(int,Uu),Uub)
        & aa(int,$o,ord_less_eq(int,Uub),Uua) ) ) ).

% ATP.lambda_331
tff(fact_8509_ATP_Olambda__332,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_aa(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_332
tff(fact_8510_ATP_Olambda__333,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_au(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_333
tff(fact_8511_ATP_Olambda__334,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_at(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_334
tff(fact_8512_ATP_Olambda__335,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aan(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_335
tff(fact_8513_ATP_Olambda__336,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_336
tff(fact_8514_ATP_Olambda__337,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,$o),aTP_Lamp_bt(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_337
tff(fact_8515_ATP_Olambda__338,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,$o),aTP_Lamp_lc(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_338
tff(fact_8516_ATP_Olambda__339,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,$o),aTP_Lamp_bv(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_339
tff(fact_8517_ATP_Olambda__340,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,$o),aTP_Lamp_ld(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_340
tff(fact_8518_ATP_Olambda__341,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aaw(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ~ aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_341
tff(fact_8519_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_iu(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_342
tff(fact_8520_ATP_Olambda__343,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_dt(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_343
tff(fact_8521_ATP_Olambda__344,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_me(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_344
tff(fact_8522_ATP_Olambda__345,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kg(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_345
tff(fact_8523_ATP_Olambda__346,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_uy(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_346
tff(fact_8524_ATP_Olambda__347,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kl(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_347
tff(fact_8525_ATP_Olambda__348,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ma(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_348
tff(fact_8526_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ly(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_349
tff(fact_8527_ATP_Olambda__350,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_lz(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_350
tff(fact_8528_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_lx(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_351
tff(fact_8529_ATP_Olambda__352,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zh(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_352
tff(fact_8530_ATP_Olambda__353,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_ab(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_353
tff(fact_8531_ATP_Olambda__354,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_kr(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_354
tff(fact_8532_ATP_Olambda__355,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_aab(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,aa(nat,A,nth(A,Uua),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_355
tff(fact_8533_ATP_Olambda__356,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_de(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_356
tff(fact_8534_ATP_Olambda__357,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_dg(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_357
tff(fact_8535_ATP_Olambda__358,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ue(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),aa(nat,int,Uua,Uub))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) ).

% ATP.lambda_358
tff(fact_8536_ATP_Olambda__359,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_od(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_359
tff(fact_8537_ATP_Olambda__360,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_di(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_360
tff(fact_8538_ATP_Olambda__361,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hr(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_361
tff(fact_8539_ATP_Olambda__362,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_so(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),Uub)) ) ).

% ATP.lambda_362
tff(fact_8540_ATP_Olambda__363,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_gz(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_363
tff(fact_8541_ATP_Olambda__364,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_da(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_364
tff(fact_8542_ATP_Olambda__365,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_df(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_365
tff(fact_8543_ATP_Olambda__366,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_fg(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_366
tff(fact_8544_ATP_Olambda__367,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_he(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_367
tff(fact_8545_ATP_Olambda__368,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(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_368
tff(fact_8546_ATP_Olambda__369,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aj(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,ord_less(nat,Uub),Uua) ) ) ).

% ATP.lambda_369
tff(fact_8547_ATP_Olambda__370,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_zz(fun(A,B),fun(list(A),fun(A,$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),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% ATP.lambda_370
tff(fact_8548_ATP_Olambda__371,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_aaa(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uu,aa(nat,A,nth(A,Uua),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% ATP.lambda_371
tff(fact_8549_ATP_Olambda__372,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oh(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_372
tff(fact_8550_ATP_Olambda__373,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wx(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = divide_divide(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_373
tff(fact_8551_ATP_Olambda__374,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_br(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_374
tff(fact_8552_ATP_Olambda__375,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ph(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_375
tff(fact_8553_ATP_Olambda__376,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ou(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_376
tff(fact_8554_ATP_Olambda__377,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_377
tff(fact_8555_ATP_Olambda__378,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_mz(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,Uu,Uub),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_378
tff(fact_8556_ATP_Olambda__379,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_379
tff(fact_8557_ATP_Olambda__380,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vi(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = divide_divide(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_380
tff(fact_8558_ATP_Olambda__381,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_381
tff(fact_8559_ATP_Olambda__382,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ro(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_382
tff(fact_8560_ATP_Olambda__383,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vw(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = divide_divide(real,aa(real,real,Uua,Uub),aa(real,real,Uu,Uub)) ).

% ATP.lambda_383
tff(fact_8561_ATP_Olambda__384,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_yc(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_384
tff(fact_8562_ATP_Olambda__385,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_ft(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_385
tff(fact_8563_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xc(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_386
tff(fact_8564_ATP_Olambda__387,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_fs(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_387
tff(fact_8565_ATP_Olambda__388,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_bq(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_388
tff(fact_8566_ATP_Olambda__389,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oo(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_389
tff(fact_8567_ATP_Olambda__390,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_390
tff(fact_8568_ATP_Olambda__391,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_391
tff(fact_8569_ATP_Olambda__392,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_ne(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_392
tff(fact_8570_ATP_Olambda__393,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_393
tff(fact_8571_ATP_Olambda__394,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_394
tff(fact_8572_ATP_Olambda__395,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ve(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_395
tff(fact_8573_ATP_Olambda__396,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_396
tff(fact_8574_ATP_Olambda__397,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_397
tff(fact_8575_ATP_Olambda__398,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qz(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_398
tff(fact_8576_ATP_Olambda__399,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_399
tff(fact_8577_ATP_Olambda__400,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vk(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_400
tff(fact_8578_ATP_Olambda__401,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_fj(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_401
tff(fact_8579_ATP_Olambda__402,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ti(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_402
tff(fact_8580_ATP_Olambda__403,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_403
tff(fact_8581_ATP_Olambda__404,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_fd(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_404
tff(fact_8582_ATP_Olambda__405,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_405
tff(fact_8583_ATP_Olambda__406,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_xs(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_406
tff(fact_8584_ATP_Olambda__407,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_qo(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_407
tff(fact_8585_ATP_Olambda__408,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_qp(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_408
tff(fact_8586_ATP_Olambda__409,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_cw(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_409
tff(fact_8587_ATP_Olambda__410,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_er(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_410
tff(fact_8588_ATP_Olambda__411,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ol(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_411
tff(fact_8589_ATP_Olambda__412,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xq(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_412
tff(fact_8590_ATP_Olambda__413,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_nf(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_413
tff(fact_8591_ATP_Olambda__414,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rf(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_414
tff(fact_8592_ATP_Olambda__415,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vh(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_415
tff(fact_8593_ATP_Olambda__416,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_re(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_416
tff(fact_8594_ATP_Olambda__417,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_le(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_417
tff(fact_8595_ATP_Olambda__418,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_nr(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_418
tff(fact_8596_ATP_Olambda__419,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pn(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_419
tff(fact_8597_ATP_Olambda__420,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xz(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_420
tff(fact_8598_ATP_Olambda__421,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_421
tff(fact_8599_ATP_Olambda__422,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_422
tff(fact_8600_ATP_Olambda__423,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_423
tff(fact_8601_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_to(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_424
tff(fact_8602_ATP_Olambda__425,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qs(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_425
tff(fact_8603_ATP_Olambda__426,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_zt(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_426
tff(fact_8604_ATP_Olambda__427,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zv(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_427
tff(fact_8605_ATP_Olambda__428,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_428
tff(fact_8606_ATP_Olambda__429,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_nj(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_429
tff(fact_8607_ATP_Olambda__430,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_rj(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),Uua,Uub),Uu) ) ).

% ATP.lambda_430
tff(fact_8608_ATP_Olambda__431,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_rl(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_431
tff(fact_8609_ATP_Olambda__432,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,real,aa(B,fun(A,real),aTP_Lamp_uz(fun(A,B),fun(B,fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_432
tff(fact_8610_ATP_Olambda__433,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_433
tff(fact_8611_ATP_Olambda__434,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_434
tff(fact_8612_ATP_Olambda__435,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_bc(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_435
tff(fact_8613_ATP_Olambda__436,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_cs(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ).

% ATP.lambda_436
tff(fact_8614_ATP_Olambda__437,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_em(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_437
tff(fact_8615_ATP_Olambda__438,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_ng(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_438
tff(fact_8616_ATP_Olambda__439,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_rn(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_439
tff(fact_8617_ATP_Olambda__440,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_su(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu) ) ).

% ATP.lambda_440
tff(fact_8618_ATP_Olambda__441,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_wc(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,ord_less(A,aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_441
tff(fact_8619_ATP_Olambda__442,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_442
tff(fact_8620_ATP_Olambda__443,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_we(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_443
tff(fact_8621_ATP_Olambda__444,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aas(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_444
tff(fact_8622_ATP_Olambda__445,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cx(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_445
tff(fact_8623_ATP_Olambda__446,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_eo(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_446
tff(fact_8624_ATP_Olambda__447,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_om(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_447
tff(fact_8625_ATP_Olambda__448,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xi(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_448
tff(fact_8626_ATP_Olambda__449,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_449
tff(fact_8627_ATP_Olambda__450,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_na(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_450
tff(fact_8628_ATP_Olambda__451,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ra(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_451
tff(fact_8629_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_dm(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_452
tff(fact_8630_ATP_Olambda__453,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qu(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_453
tff(fact_8631_ATP_Olambda__454,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_454
tff(fact_8632_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pa(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu) ) ).

% ATP.lambda_455
tff(fact_8633_ATP_Olambda__456,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_du(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_456
tff(fact_8634_ATP_Olambda__457,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_st(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_457
tff(fact_8635_ATP_Olambda__458,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qd(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_458
tff(fact_8636_ATP_Olambda__459,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ge(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_459
tff(fact_8637_ATP_Olambda__460,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ru(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_460
tff(fact_8638_ATP_Olambda__461,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_nn(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_461
tff(fact_8639_ATP_Olambda__462,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_bs(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_462
tff(fact_8640_ATP_Olambda__463,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_pd(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_463
tff(fact_8641_ATP_Olambda__464,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_xr(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_464
tff(fact_8642_ATP_Olambda__465,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_nk(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_465
tff(fact_8643_ATP_Olambda__466,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_466
tff(fact_8644_ATP_Olambda__467,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_vo(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_467
tff(fact_8645_ATP_Olambda__468,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_qr(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_468
tff(fact_8646_ATP_Olambda__469,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_qn(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_469
tff(fact_8647_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vf(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_470
tff(fact_8648_ATP_Olambda__471,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qf(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_471
tff(fact_8649_ATP_Olambda__472,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_xa(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_472
tff(fact_8650_ATP_Olambda__473,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_mf(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_473
tff(fact_8651_ATP_Olambda__474,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_nq(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_474
tff(fact_8652_ATP_Olambda__475,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vl(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_475
tff(fact_8653_ATP_Olambda__476,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aac(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_476
tff(fact_8654_ATP_Olambda__477,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cn(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = 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_477
tff(fact_8655_ATP_Olambda__478,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_dj(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_478
tff(fact_8656_ATP_Olambda__479,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_xd(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_479
tff(fact_8657_ATP_Olambda__480,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_jn(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_480
tff(fact_8658_ATP_Olambda__481,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wt(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_481
tff(fact_8659_ATP_Olambda__482,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_ls(nat,fun(nat,fun(word(A),$o)),Uu),Uua),Uub)
        <=> aa(word(A),$o,ord_less(word(A),Uub),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_482
tff(fact_8660_ATP_Olambda__483,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_lt(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,Uub),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_483
tff(fact_8661_ATP_Olambda__484,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_gg(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_484
tff(fact_8662_ATP_Olambda__485,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_fm(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_485
tff(fact_8663_ATP_Olambda__486,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_gf(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_486
tff(fact_8664_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fr(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_487
tff(fact_8665_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fp(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_488
tff(fact_8666_ATP_Olambda__489,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_lm(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_489
tff(fact_8667_ATP_Olambda__490,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_gm(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_490
tff(fact_8668_ATP_Olambda__491,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ts(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = divide_divide(real,Uua,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_491
tff(fact_8669_ATP_Olambda__492,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_ew(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_492
tff(fact_8670_ATP_Olambda__493,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fq(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_493
tff(fact_8671_ATP_Olambda__494,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_ks(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),replicate(A,Uu,Uua)) ) ).

% ATP.lambda_494
tff(fact_8672_ATP_Olambda__495,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_aam(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),cons(A,Uub),map_filter(B,A,Uu,Uua)) ).

% ATP.lambda_495
tff(fact_8673_ATP_Olambda__496,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_wo(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_496
tff(fact_8674_ATP_Olambda__497,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_wg(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_497
tff(fact_8675_ATP_Olambda__498,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wi(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_498
tff(fact_8676_ATP_Olambda__499,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_mm(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,Uua,aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_499
tff(fact_8677_ATP_Olambda__500,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_wb(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,ord_less(A,Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_500
tff(fact_8678_ATP_Olambda__501,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_501
tff(fact_8679_ATP_Olambda__502,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_502
tff(fact_8680_ATP_Olambda__503,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aat(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_503
tff(fact_8681_ATP_Olambda__504,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ky(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_504
tff(fact_8682_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xj(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_505
tff(fact_8683_ATP_Olambda__506,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_cr(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_506
tff(fact_8684_ATP_Olambda__507,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_en(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_507
tff(fact_8685_ATP_Olambda__508,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_dv(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_508
tff(fact_8686_ATP_Olambda__509,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ss(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_509
tff(fact_8687_ATP_Olambda__510,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qe(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_510
tff(fact_8688_ATP_Olambda__511,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cy(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_511
tff(fact_8689_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_on(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_512
tff(fact_8690_ATP_Olambda__513,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xu(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_513
tff(fact_8691_ATP_Olambda__514,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_nb(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_514
tff(fact_8692_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_rb(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_515
tff(fact_8693_ATP_Olambda__516,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_dl(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_516
tff(fact_8694_ATP_Olambda__517,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_517
tff(fact_8695_ATP_Olambda__518,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qy(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_518
tff(fact_8696_ATP_Olambda__519,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vd(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_519
tff(fact_8697_ATP_Olambda__520,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_fx(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_520
tff(fact_8698_ATP_Olambda__521,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ty(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_521
tff(fact_8699_ATP_Olambda__522,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rg(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_522
tff(fact_8700_ATP_Olambda__523,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_xx(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_523
tff(fact_8701_ATP_Olambda__524,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_sw(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_524
tff(fact_8702_ATP_Olambda__525,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_rh(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_525
tff(fact_8703_ATP_Olambda__526,axiom,
    ! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_dq(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_526
tff(fact_8704_ATP_Olambda__527,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bj(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_527
tff(fact_8705_ATP_Olambda__528,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_wu(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_528
tff(fact_8706_ATP_Olambda__529,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_aap(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),Uu) ).

% ATP.lambda_529
tff(fact_8707_ATP_Olambda__530,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ba(fun(A,nat),fun(A,fun(A,$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_530
tff(fact_8708_ATP_Olambda__531,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bl(fun(A,nat),fun(fun(A,$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_531
tff(fact_8709_ATP_Olambda__532,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_az(fun(A,nat),fun(fun(A,nat),fun(A,$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_532
tff(fact_8710_ATP_Olambda__533,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_fz(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_533
tff(fact_8711_ATP_Olambda__534,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_fy(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_534
tff(fact_8712_ATP_Olambda__535,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_te(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_535
tff(fact_8713_ATP_Olambda__536,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_qh(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_536
tff(fact_8714_ATP_Olambda__537,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_mx(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_537
tff(fact_8715_ATP_Olambda__538,axiom,
    ! [Uu: fun(real,$o),Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_wm(fun(real,$o),fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ) ).

% ATP.lambda_538
tff(fact_8716_ATP_Olambda__539,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_vc(fun(real,A),fun(real,fun(real,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ).

% ATP.lambda_539
tff(fact_8717_ATP_Olambda__540,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_vx(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_540
tff(fact_8718_ATP_Olambda__541,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cq(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_541
tff(fact_8719_ATP_Olambda__542,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sv(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_542
tff(fact_8720_ATP_Olambda__543,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xb(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_543
tff(fact_8721_ATP_Olambda__544,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_be(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_544
tff(fact_8722_ATP_Olambda__545,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_eu(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_545
tff(fact_8723_ATP_Olambda__546,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_wf(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_546
tff(fact_8724_ATP_Olambda__547,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_qi(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_547
tff(fact_8725_ATP_Olambda__548,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_si(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_548
tff(fact_8726_ATP_Olambda__549,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_mu(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_549
tff(fact_8727_ATP_Olambda__550,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_zs(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_550
tff(fact_8728_ATP_Olambda__551,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ux(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),Uub)) ) ).

% ATP.lambda_551
tff(fact_8729_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qg(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_552
tff(fact_8730_ATP_Olambda__553,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_dx(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_553
tff(fact_8731_ATP_Olambda__554,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_oz(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_554
tff(fact_8732_ATP_Olambda__555,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_yw(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_555
tff(fact_8733_ATP_Olambda__556,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_xf(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_556
tff(fact_8734_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_xg(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8735_ATP_Olambda__558,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ul(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_558
tff(fact_8736_ATP_Olambda__559,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_mr(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_559
tff(fact_8737_ATP_Olambda__560,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_qk(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_560
tff(fact_8738_ATP_Olambda__561,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_ms(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_561
tff(fact_8739_ATP_Olambda__562,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_va(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_562
tff(fact_8740_ATP_Olambda__563,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_jx(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_563
tff(fact_8741_ATP_Olambda__564,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_jw(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_564
tff(fact_8742_ATP_Olambda__565,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abj(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_565
tff(fact_8743_ATP_Olambda__566,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_sq(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_sp(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_566
tff(fact_8744_ATP_Olambda__567,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qc(fun(nat,A),fun(A,fun(A,A)),Uu),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_qb(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_567
tff(fact_8745_ATP_Olambda__568,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_db(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_568
tff(fact_8746_ATP_Olambda__569,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_gh(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_569
tff(fact_8747_ATP_Olambda__570,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ud(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_570
tff(fact_8748_ATP_Olambda__571,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_ar(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_high(Uu,Uua)) ).

% ATP.lambda_571
tff(fact_8749_ATP_Olambda__572,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_aq(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_low(Uu,Uua)) ).

% ATP.lambda_572
tff(fact_8750_ATP_Olambda__573,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_abg(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = finite_card(B,collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_abf(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_573
tff(fact_8751_ATP_Olambda__574,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_pz(fun(B,real),fun(fun(real,A),fun(B,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu,Uub)))) ) ).

% ATP.lambda_574
tff(fact_8752_ATP_Olambda__575,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_wy(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N6: nat] :
              ( aa(nat,$o,ord_less_eq(nat,Uub),N6)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,Uub,N6)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_575
tff(fact_8753_ATP_Olambda__576,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_xe(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A7: nat] :
              ( aa(nat,$o,ord_less_eq(nat,Uub),A7)
             => ! [B7: nat] :
                  ( aa(nat,$o,ord_less(nat,A7),B7)
                 => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or3652927894154168847AtMost(nat,A7,B7)))),aa(nat,real,Uua,A7)) ) ) ) ) ).

% ATP.lambda_576
tff(fact_8754_ATP_Olambda__577,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_zl(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X5: A,Xs4: list(A)] :
          ( ( Uub = aa(list(A),list(A),cons(A,X5),Xs4) )
          & member(A,X5,Uu)
          & member(list(A),Xs4,Uua) ) ) ).

% ATP.lambda_577
tff(fact_8755_ATP_Olambda__578,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_il(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,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))),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_578
tff(fact_8756_ATP_Olambda__579,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_ih(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),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))),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_579
tff(fact_8757_ATP_Olambda__580,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_ij(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,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))),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_580
tff(fact_8758_ATP_Olambda__581,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_hv(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_581
tff(fact_8759_ATP_Olambda__582,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_hx(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_582
tff(fact_8760_ATP_Olambda__583,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_hy(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_583
tff(fact_8761_ATP_Olambda__584,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_hw(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_584
tff(fact_8762_ATP_Olambda__585,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_fe(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_585
tff(fact_8763_ATP_Olambda__586,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_ca(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_586
tff(fact_8764_ATP_Olambda__587,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_fi(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_587
tff(fact_8765_ATP_Olambda__588,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_abm(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_588
tff(fact_8766_ATP_Olambda__589,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_kz(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_589
tff(fact_8767_ATP_Olambda__590,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_gu(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_gt(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_590
tff(fact_8768_ATP_Olambda__591,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_eq(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_ep(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_591
tff(fact_8769_ATP_Olambda__592,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_hz(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_592
tff(fact_8770_ATP_Olambda__593,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_ht(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_hs(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_593
tff(fact_8771_ATP_Olambda__594,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_ho(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_hn(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_594
tff(fact_8772_ATP_Olambda__595,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_nz(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_595
tff(fact_8773_ATP_Olambda__596,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_nx(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_596
tff(fact_8774_ATP_Olambda__597,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_nw(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_597
tff(fact_8775_ATP_Olambda__598,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_nv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_598
tff(fact_8776_ATP_Olambda__599,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_gi(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_599
tff(fact_8777_ATP_Olambda__600,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_gt(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_600
tff(fact_8778_ATP_Olambda__601,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ua(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua)),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_601
tff(fact_8779_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_wz(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_602
tff(fact_8780_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_wp(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_603
tff(fact_8781_ATP_Olambda__604,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_hq(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_604
tff(fact_8782_ATP_Olambda__605,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_hm(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_605
tff(fact_8783_ATP_Olambda__606,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_hp(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_606
tff(fact_8784_ATP_Olambda__607,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_jl(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_607
tff(fact_8785_ATP_Olambda__608,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_gk(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_608
tff(fact_8786_ATP_Olambda__609,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_abf(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_609
tff(fact_8787_ATP_Olambda__610,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_abe(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_610
tff(fact_8788_ATP_Olambda__611,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_gj(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_611
tff(fact_8789_ATP_Olambda__612,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_abk(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_612
tff(fact_8790_ATP_Olambda__613,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_gl(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_613
tff(fact_8791_ATP_Olambda__614,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_hg(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_614
tff(fact_8792_ATP_Olambda__615,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_bw(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_615
tff(fact_8793_ATP_Olambda__616,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_bu(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_616
tff(fact_8794_ATP_Olambda__617,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_dr(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_617
tff(fact_8795_ATP_Olambda__618,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ui(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,divide_divide(real,one_one(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_618
tff(fact_8796_ATP_Olambda__619,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_dh(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_619
tff(fact_8797_ATP_Olambda__620,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_lg(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_620
tff(fact_8798_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_qb(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,minus_minus(A,divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_621
tff(fact_8799_ATP_Olambda__622,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_sp(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_622
tff(fact_8800_ATP_Olambda__623,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu,Uua))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_623
tff(fact_8801_ATP_Olambda__624,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_if(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_624
tff(fact_8802_ATP_Olambda__625,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_lh(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_625
tff(fact_8803_ATP_Olambda__626,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_hs(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_626
tff(fact_8804_ATP_Olambda__627,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_hk(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_627
tff(fact_8805_ATP_Olambda__628,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_hn(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_628
tff(fact_8806_ATP_Olambda__629,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_ep(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_629
tff(fact_8807_ATP_Olambda__630,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pu(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_630
tff(fact_8808_ATP_Olambda__631,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ps(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_631
tff(fact_8809_ATP_Olambda__632,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_or(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,exp(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_632
tff(fact_8810_ATP_Olambda__633,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ot(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_633
tff(fact_8811_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_634
tff(fact_8812_ATP_Olambda__635,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_og(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_635
tff(fact_8813_ATP_Olambda__636,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_pc(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),sin(real,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_636
tff(fact_8814_ATP_Olambda__637,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_py(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_637
tff(fact_8815_ATP_Olambda__638,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_uq(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_638
tff(fact_8816_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_um(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub))) ) ).

% ATP.lambda_639
tff(fact_8817_ATP_Olambda__640,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_ws(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub)) ) ) ).

% ATP.lambda_640
tff(fact_8818_ATP_Olambda__641,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_us(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ur(A,A)))))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ur(A,A))))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ur(A,A)))))) ) ).

% ATP.lambda_641
tff(fact_8819_ATP_Olambda__642,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_uo(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uua)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,minus_minus(A,Uuc),Uua)))) ) ).

% ATP.lambda_642
tff(fact_8820_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_un(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))) ) ).

% ATP.lambda_643
tff(fact_8821_ATP_Olambda__644,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_abl(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_abk(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_644
tff(fact_8822_ATP_Olambda__645,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_za(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),cons(A,Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_645
tff(fact_8823_ATP_Olambda__646,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_do(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_646
tff(fact_8824_ATP_Olambda__647,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_vt(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less_eq(real,Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_da(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_647
tff(fact_8825_ATP_Olambda__648,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_zp(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = ord_min(A,Uu,min_list(A,Uua)) ) ).

% ATP.lambda_648
tff(fact_8826_ATP_Olambda__649,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_zm(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_649
tff(fact_8827_ATP_Olambda__650,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_cj(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_650
tff(fact_8828_ATP_Olambda__651,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_ez(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_651
tff(fact_8829_ATP_Olambda__652,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_bf(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_652
tff(fact_8830_ATP_Olambda__653,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_ex(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_653
tff(fact_8831_ATP_Olambda__654,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_aav(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_aau(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_654
tff(fact_8832_ATP_Olambda__655,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_ox(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_655
tff(fact_8833_ATP_Olambda__656,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_pm(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_pl(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_656
tff(fact_8834_ATP_Olambda__657,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_ic(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_ib(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_657
tff(fact_8835_ATP_Olambda__658,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_dn(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_658
tff(fact_8836_ATP_Olambda__659,axiom,
    ! [Uu: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_ny(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_nx(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,minus_minus(nat,Uu),Uuc)),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu),Uuc)))))) ).

% ATP.lambda_659
tff(fact_8837_ATP_Olambda__660,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_ig(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_if(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_660
tff(fact_8838_ATP_Olambda__661,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_je(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_661
tff(fact_8839_ATP_Olambda__662,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_jb(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_662
tff(fact_8840_ATP_Olambda__663,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_jc(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_663
tff(fact_8841_ATP_Olambda__664,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_ib(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_664
tff(fact_8842_ATP_Olambda__665,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: nat,Uud: A] : aa(A,B,aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_pe(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(nat,nat,minus_minus(nat,Uuc),one_one(nat)))) ) ).

% ATP.lambda_665
tff(fact_8843_ATP_Olambda__666,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_jr(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_666
tff(fact_8844_ATP_Olambda__667,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_dp(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_667
tff(fact_8845_ATP_Olambda__668,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_ju(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_668
tff(fact_8846_ATP_Olambda__669,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_pl(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_pj(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),aa(set(A),set(A),minus_minus(set(A),Uu),insert(A,Uue,bot_bot(set(A)))))) ) ).

% ATP.lambda_669
tff(fact_8847_ATP_Olambda__670,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ov(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = divide_divide(B,aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_670
tff(fact_8848_ATP_Olambda__671,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_po(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu,Uub)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub)),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_671
tff(fact_8849_ATP_Olambda__672,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_oi(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_672
tff(fact_8850_ATP_Olambda__673,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_op(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_673
tff(fact_8851_ATP_Olambda__674,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pi(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),divide_divide(B,aa(A,B,Uua,Uue),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_674
tff(fact_8852_ATP_Olambda__675,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_aau(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(
        x2: b,
        x2:= aa(a,b,Uu,Uub),
        $ite(
          aa(b,$o,ord_less(b,x2),Uua),
          aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),product_Pair(list(a),product_prod(list(a),list(a)),aa(list(a),list(a),cons(a,Uub),Uuc)),aa(list(a),product_prod(list(a),list(a)),product_Pair(list(a),list(a),Uud),Uue)),
          $ite(aa(b,$o,ord_less(b,Uua),x2),aa(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)),product_Pair(list(a),list(a),Uud),aa(list(a),list(a),cons(a,Uub),Uue))),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),product_Pair(list(a),product_prod(list(a),list(a)),Uuc),aa(list(a),product_prod(list(a),list(a)),product_Pair(list(a),list(a),aa(list(a),list(a),cons(a,Uub),Uud)),Uue))) ) ) ).

% ATP.lambda_675
tff(fact_8853_ATP_Olambda__676,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_yk(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,ord_less_eq(nat,Uue),Uuf)
        & aa(nat,$o,ord_less(nat,Uuf),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
        & ! [I2: nat] :
            ( aa(nat,$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_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I2)),X_13)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I2) ) )
        & $ite(
            Uue = Uuf,
            ! [X5: vEBT_VEBT] :
              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X8) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X5: nat] :
                ( aa(nat,$o,ord_less(nat,X5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X5)
                 => ( aa(nat,$o,ord_less(nat,Uue),X5)
                    & aa(nat,$o,ord_less_eq(nat,X5),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_676
tff(fact_8854_ATP_Olambda__677,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_kt(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uuf),aa(B,assn,aa(A,fun(B,assn),Uu,aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Uua),Uub)),Uue)),aa(nat,B,nth(B,aa(list(B),list(B),cons(B,Uuc),Uud)),Uue))) ).

% ATP.lambda_677
tff(fact_8855_ATP_Olambda__678,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_kx(assn,fun(A,assn),Uu),Uua) = Uu ).

% ATP.lambda_678
tff(fact_8856_ATP_Olambda__679,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_oj(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_679
tff(fact_8857_ATP_Olambda__680,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aad(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_680
tff(fact_8858_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cv(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_681
tff(fact_8859_ATP_Olambda__682,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ni(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_682
tff(fact_8860_ATP_Olambda__683,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aay(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_683
tff(fact_8861_ATP_Olambda__684,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_aal(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_684
tff(fact_8862_ATP_Olambda__685,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aaz(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_685
tff(fact_8863_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_qj(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_686
tff(fact_8864_ATP_Olambda__687,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_aaq(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_687
tff(fact_8865_ATP_Olambda__688,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lk(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_688
tff(fact_8866_ATP_Olambda__689,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_cf(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_689
tff(fact_8867_ATP_Olambda__690,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aar(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_690
tff(fact_8868_ATP_Olambda__691,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_zq(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_691
tff(fact_8869_ATP_Olambda__692,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_zr(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_692
tff(fact_8870_ATP_Olambda__693,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_ga(complex,complex),Uu) = Uu ).

% ATP.lambda_693
tff(fact_8871_ATP_Olambda__694,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_bg(nat,nat),Uu) = Uu ).

% ATP.lambda_694
tff(fact_8872_ATP_Olambda__695,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_bp(int,int),Uu) = Uu ).

% ATP.lambda_695
tff(fact_8873_ATP_Olambda__696,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ur(A,A),Uu) = Uu ) ).

% ATP.lambda_696
tff(fact_8874_ATP_Olambda__697,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_nh(A,A),Uu) = Uu ) ).

% ATP.lambda_697
tff(fact_8875_ATP_Olambda__698,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ut(A,A),Uu) = Uu ) ).

% ATP.lambda_698
tff(fact_8876_ATP_Olambda__699,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_uu(A,A),Uu) = Uu ) ).

% ATP.lambda_699
tff(fact_8877_ATP_Olambda__700,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_mp(A,A),Uu) = Uu ) ).

% ATP.lambda_700
tff(fact_8878_ATP_Olambda__701,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_af(A,A),Uu) = Uu ) ).

% ATP.lambda_701
tff(fact_8879_ATP_Olambda__702,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_cc(A,A),Uu) = Uu ).

% ATP.lambda_702
tff(fact_8880_ATP_Olambda__703,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_cp(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_703
tff(fact_8881_ATP_Olambda__704,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ch(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_704
tff(fact_8882_ATP_Olambda__705,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_eg(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_705
tff(fact_8883_ATP_Olambda__706,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ok(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_706
tff(fact_8884_ATP_Olambda__707,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ag(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_707
tff(fact_8885_ATP_Olambda__708,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_bo(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_708
tff(fact_8886_ATP_Olambda__709,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_abh(A,real),Uu) = one_one(real) ).

% ATP.lambda_709
tff(fact_8887_ATP_Olambda__710,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_abc(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_710
tff(fact_8888_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_la(B,option(A)),Uu) = none(A) ).

% ATP.lambda_711
tff(fact_8889_ATP_Olambda__712,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_aw(A,option(B)),Uu) = none(B) ).

% ATP.lambda_712
tff(fact_8890_ATP_Olambda__713,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_km(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_713
tff(fact_8891_ATP_Olambda__714,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_mh(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_714
tff(fact_8892_ATP_Olambda__715,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_yl(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_715
tff(fact_8893_ATP_Olambda__716,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_mj(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_716
tff(fact_8894_ATP_Olambda__717,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_ym(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_717
tff(fact_8895_ATP_Olambda__718,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_ax(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_718

% Type constructors (1297)
tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
    enum(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_1,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( enum(A9)
        & enum(A10) )
     => enum(product_prod(A9,A10)) ) ).

tff(tcon_Numeral__Type_Onum1___Enum_Oenum_2,axiom,
    enum(numeral_num1) ).

tff(tcon_Numeral__Type_Obit1___Enum_Oenum_3,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => enum(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Enum_Oenum_4,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => enum(numeral_bit0(A9)) ) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_5,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Option_Ooption___Enum_Oenum_6,axiom,
    ! [A9: $tType] :
      ( enum(A9)
     => enum(option(A9)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_7,axiom,
    ! [A9: $tType] :
      ( comple5582772986160207858norder(A9)
     => comple5582772986160207858norder(option(A9)) ) ).

tff(tcon_Enum_Ofinite__3___Enum_Oenum_8,axiom,
    enum(finite_3) ).

tff(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_9,axiom,
    comple5582772986160207858norder(finite_3) ).

tff(tcon_Enum_Ofinite__2___Enum_Oenum_10,axiom,
    enum(finite_2) ).

tff(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_11,axiom,
    comple5582772986160207858norder(finite_2) ).

tff(tcon_Enum_Ofinite__1___Enum_Oenum_12,axiom,
    enum(finite_1) ).

tff(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_13,axiom,
    comple5582772986160207858norder(finite_1) ).

tff(tcon_Sum__Type_Osum___Enum_Oenum_14,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( enum(A9)
        & enum(A10) )
     => enum(sum_sum(A9,A10)) ) ).

tff(tcon_String_Ochar___Enum_Oenum_15,axiom,
    enum(char) ).

tff(tcon_Word_Oword___Enum_Oenum_16,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => enum(word(A9)) ) ).

tff(tcon_HOL_Obool___Enum_Oenum_17,axiom,
    enum($o) ).

tff(tcon_Set_Oset___Enum_Oenum_18,axiom,
    ! [A9: $tType] :
      ( enum(A9)
     => enum(set(A9)) ) ).

tff(tcon_fun___Enum_Oenum_19,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( enum(A9)
        & enum(A10) )
     => enum(fun(A9,A10)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A9: $tType,A10: $tType] :
      ( boolea8198339166811842893lgebra(A10)
     => boolea8198339166811842893lgebra(fun(A9,A10)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A9: $tType,A10: $tType] :
      ( order_top(A10)
     => order_top(fun(A9,A10)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A9: $tType,A10: $tType] :
      ( order_bot(A10)
     => order_bot(fun(A9,A10)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A9: $tType,A10: $tType] :
      ( preorder(A10)
     => preorder(fun(A9,A10)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A10) )
     => finite_finite(fun(A9,A10)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A9: $tType,A10: $tType] :
      ( order(A10)
     => order(fun(A9,A10)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ord(A10)
     => ord(fun(A9,A10)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A9: $tType,A10: $tType] :
      ( uminus(A10)
     => uminus(fun(A9,A10)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A9: $tType,A10: $tType] :
      ( minus(A10)
     => minus(fun(A9,A10)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( enum(A9)
        & cl_HOL_Oequal(A10) )
     => cl_HOL_Oequal(fun(A9,A10)) ) ).

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___Limits_Otopological__comm__monoid__mult,axiom,
    topolo4987421752381908075d_mult(int) ).

tff(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
    bit_bi6583157726757044596ension(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
    euclid5891614535332579305n_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
    topolo1898628316856586783d_mult(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___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___Limits_Otopological__monoid__add,axiom,
    topolo6943815403480290642id_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___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___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_20,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_21,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_22,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_23,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_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___Groups_Ominus_24,axiom,
    minus(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_25,axiom,
    cl_HOL_Oequal(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_26,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_27,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_28,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_29,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_30,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_31,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_32,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_33,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_34,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_35,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_36,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_37,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_38,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_39,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_40,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_41,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_42,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_43,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_44,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_45,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_46,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_47,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_48,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_49,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_50,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_51,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_52,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_53,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_54,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_55,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_56,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_57,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_58,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_59,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_60,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_61,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_62,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_63,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_64,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_65,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_66,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_67,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_68,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_69,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_70,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_71,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_72,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_73,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_74,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_75,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_76,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_77,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_78,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_79,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_80,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_81,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_82,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_83,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_84,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_85,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_86,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_87,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_88,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_89,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_90,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_91,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_92,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_93,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_94,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_95,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_96,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_97,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_98,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_99,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_100,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_101,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_102,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_103,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_104,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_105,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_106,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_107,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_108,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_109,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_110,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_111,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_112,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_113,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_114,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_115,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_116,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_117,axiom,
    plus(num) ).

tff(tcon_Num_Onum___HOL_Oequal_118,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Num_Onum___Nat_Osize_119,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_120,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_121,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_122,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_123,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_124,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_125,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_126,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_127,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_128,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_129,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_130,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_131,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_132,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_133,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_134,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_135,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_136,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_137,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_138,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_139,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_140,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_141,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_142,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_143,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_144,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_145,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_146,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_147,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_148,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_149,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_150,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_151,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_152,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_153,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_154,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_155,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_156,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_157,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_158,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_159,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_160,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_161,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_162,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_163,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_164,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_165,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_166,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_167,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_168,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_169,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_170,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_171,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_172,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_173,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_174,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_175,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_176,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_177,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_178,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_179,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_180,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_181,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_182,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_183,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_184,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_185,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_186,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_187,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_188,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_189,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_190,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_191,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_192,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_193,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_194,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_195,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_196,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_197,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_198,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_199,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_200,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_201,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_202,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_203,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_204,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_205,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_206,axiom,
    ! [A9: $tType] : boolea8198339166811842893lgebra(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_207,axiom,
    ! [A9: $tType] : order_top(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_208,axiom,
    ! [A9: $tType] : order_bot(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_209,axiom,
    ! [A9: $tType] : preorder(set(A9)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_210,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(set(A9)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder_211,axiom,
    ! [A9: $tType] : order(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oord_212,axiom,
    ! [A9: $tType] : ord(set(A9)) ).

tff(tcon_Set_Oset___Groups_Ouminus_213,axiom,
    ! [A9: $tType] : uminus(set(A9)) ).

tff(tcon_Set_Oset___Groups_Ominus_214,axiom,
    ! [A9: $tType] : minus(set(A9)) ).

tff(tcon_Set_Oset___HOL_Oequal_215,axiom,
    ! [A9: $tType] :
      ( cl_HOL_Oequal(A9)
     => cl_HOL_Oequal(set(A9)) ) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_216,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_217,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_218,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_219,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_220,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_221,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_222,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_223,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_224,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_225,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_226,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_227,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_228,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_229,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_230,axiom,
    heap($o) ).

tff(tcon_HOL_Obool___HOL_Oequal_231,axiom,
    cl_HOL_Oequal($o) ).

tff(tcon_Heap_Oref___Heap_Oheap_232,axiom,
    ! [A9: $tType] : heap(ref(A9)) ).

tff(tcon_Heap_Oref___HOL_Oequal_233,axiom,
    ! [A9: $tType] : cl_HOL_Oequal(ref(A9)) ).

tff(tcon_Heap_Oref___Nat_Osize_234,axiom,
    ! [A9: $tType] : size(ref(A9)) ).

tff(tcon_List_Olist___Heap_Oheap_235,axiom,
    ! [A9: $tType] :
      ( heap(A9)
     => heap(list(A9)) ) ).

tff(tcon_List_Olist___HOL_Oequal_236,axiom,
    ! [A9: $tType] : cl_HOL_Oequal(list(A9)) ).

tff(tcon_List_Olist___Nat_Osize_237,axiom,
    ! [A9: $tType] : size(list(A9)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_238,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_239,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_240,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
    topolo8458572112393995274pology(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
    topolo3112930676232923870pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_241,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_242,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_243,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_244,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_245,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_246,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_247,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_248,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_249,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_250,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_251,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_252,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_253,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_254,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_255,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_256,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_257,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
    real_V8037385150606011577_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_258,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
    topolo7287701948861334536_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_259,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_260,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_261,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_262,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_263,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_264,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_265,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_266,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_267,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
    real_V7773925162809079976_field(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__monoid__add_268,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_269,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_270,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_271,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_272,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_273,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_274,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_275,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_276,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_277,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_278,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_279,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_280,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_281,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_282,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_283,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_284,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_285,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_286,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_287,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_288,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_289,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_290,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_291,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_292,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_293,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_294,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_295,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_296,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_297,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_298,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_299,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_300,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_301,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_302,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_303,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_304,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_305,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_306,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_307,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_308,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_309,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_310,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_311,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_312,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_313,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_314,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_315,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_316,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_317,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_318,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_319,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_320,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_321,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_322,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_323,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_324,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_325,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_326,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_327,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_328,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_329,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_330,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Otimes_331,axiom,
    times(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_332,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_333,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_334,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_335,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_336,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_337,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_338,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_339,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_340,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_341,axiom,
    dvd(real) ).

tff(tcon_Real_Oreal___HOL_Oequal_342,axiom,
    cl_HOL_Oequal(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_343,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => bit_se359711467146920520ations(word(A9)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_344,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => bit_bi6583157726757044596ension(word(A9)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_345,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => bit_ri3973907225187159222ations(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_346,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => cancel2418104881723323429up_add(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_347,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => cancel1802427076303600483id_add(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_348,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_s4317794764714335236cancel(word(A9)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_349,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => bit_semiring_bits(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_350,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => cancel_semigroup_add(word(A9)) ) ).

tff(tcon_Word_Oword___Least__significant__bit_Olsb_351,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => least_6119777620449941438nt_lsb(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_352,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => ab_semigroup_mult(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_353,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semiring_1_cancel(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_354,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_monoid_mult(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_355,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => ab_semigroup_add(word(A9)) ) ).

tff(tcon_Word_Oword___Generic__set__bit_Oset__bit_356,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => generic_set_set_bit(word(A9)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_357,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semiring_parity(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_358,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_monoid_add(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_359,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semiring_modulo(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_360,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_semiring_1(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_361,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_semiring_0(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_362,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semigroup_mult(word(A9)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_363,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semiring_numeral(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_364,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semigroup_add(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_365,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_semiring(word(A9)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_366,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => wellorder(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_367,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => ab_group_add(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_368,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => zero_neq_one(word(A9)) ) ).

tff(tcon_Word_Oword___Parity_Oring__parity_369,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => ring_parity(word(A9)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_370,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => preorder(word(A9)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_371,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => linorder(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_372,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => monoid_mult(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_373,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_ring_1(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_374,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => monoid_add(word(A9)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_375,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => finite_finite(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_376,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semiring_1(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_377,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semiring_0(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_378,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => group_add(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_379,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => mult_zero(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_380,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => comm_ring(word(A9)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_381,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => order(word(A9)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_382,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => neg_numeral(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_383,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => semiring(word(A9)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_384,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => ord(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_385,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => uminus(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_386,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => ring_1(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Otimes_387,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => times(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ominus_388,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => minus(word(A9)) ) ).

tff(tcon_Word_Oword___Power_Opower_389,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => power(word(A9)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_390,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => numeral(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_391,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => zero(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_392,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => plus(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Oring_393,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => ring(word(A9)) ) ).

tff(tcon_Word_Oword___Groups_Oone_394,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => one(word(A9)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_395,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => dvd(word(A9)) ) ).

tff(tcon_Word_Oword___HOL_Oequal_396,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => cl_HOL_Oequal(word(A9)) ) ).

tff(tcon_Word_Oword___Nat_Osize_397,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => size(word(A9)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_398,axiom,
    ! [A9: $tType] : heap(array(A9)) ).

tff(tcon_Heap_Oarray___HOL_Oequal_399,axiom,
    ! [A9: $tType] : cl_HOL_Oequal(array(A9)) ).

tff(tcon_Heap_Oarray___Nat_Osize_400,axiom,
    ! [A9: $tType] : size(array(A9)) ).

tff(tcon_String_Ochar___Orderings_Opreorder_401,axiom,
    preorder(char) ).

tff(tcon_String_Ochar___Orderings_Olinorder_402,axiom,
    linorder(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_403,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Orderings_Oorder_404,axiom,
    order(char) ).

tff(tcon_String_Ochar___Orderings_Oord_405,axiom,
    ord(char) ).

tff(tcon_String_Ochar___Heap_Oheap_406,axiom,
    heap(char) ).

tff(tcon_String_Ochar___HOL_Oequal_407,axiom,
    cl_HOL_Oequal(char) ).

tff(tcon_String_Ochar___Nat_Osize_408,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_409,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A10) )
     => finite_finite(sum_sum(A9,A10)) ) ).

tff(tcon_Sum__Type_Osum___Heap_Oheap_410,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( heap(A9)
        & heap(A10) )
     => heap(sum_sum(A9,A10)) ) ).

tff(tcon_Sum__Type_Osum___HOL_Oequal_411,axiom,
    ! [A9: $tType,A10: $tType] : cl_HOL_Oequal(sum_sum(A9,A10)) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_412,axiom,
    ! [A9: $tType,A10: $tType] : size(sum_sum(A9,A10)) ).

tff(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_413,axiom,
    condit6923001295902523014norder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_414,axiom,
    ordere1937475149494474687imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_415,axiom,
    semiri6575147826004484403cancel(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_416,axiom,
    strict9044650504122735259up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_417,axiom,
    ordere1170586879665033532d_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_418,axiom,
    ordere580206878836729694up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_419,axiom,
    ordere2412721322843649153imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_420,axiom,
    linord2810124833399127020strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_421,axiom,
    strict7427464778891057005id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_422,axiom,
    ordere8940638589300402666id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_423,axiom,
    canoni5634975068530333245id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_424,axiom,
    linord4140545234300271783up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_425,axiom,
    linord8928482502909563296strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_426,axiom,
    boolea8198339166811842893lgebra(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_427,axiom,
    semiri3467727345109120633visors(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_428,axiom,
    ordere6658533253407199908up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_429,axiom,
    ordere166539214618696060dd_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_430,axiom,
    ordere6911136660526730532id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_431,axiom,
    linord5086331880401160121up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_432,axiom,
    cancel2418104881723323429up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_433,axiom,
    cancel1802427076303600483id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_434,axiom,
    linord4710134922213307826strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_435,axiom,
    ordere2520102378445227354miring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_436,axiom,
    ordered_ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_437,axiom,
    cancel_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_438,axiom,
    linordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_439,axiom,
    ordered_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_440,axiom,
    dense_linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_441,axiom,
    ab_semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_442,axiom,
    comm_monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_443,axiom,
    comm_monoid_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_444,axiom,
    ab_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_445,axiom,
    ordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_446,axiom,
    ordered_ring_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_447,axiom,
    comm_monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_448,axiom,
    linordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_449,axiom,
    comm_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__order_450,axiom,
    dense_order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_451,axiom,
    semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_452,axiom,
    semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_453,axiom,
    comm_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Owellorder_454,axiom,
    wellorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__top_455,axiom,
    order_top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_456,axiom,
    order_bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__group__add_457,axiom,
    ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring_458,axiom,
    ordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Opreorder_459,axiom,
    preorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Olinorder_460,axiom,
    linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_461,axiom,
    monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__add_462,axiom,
    monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_463,axiom,
    finite_finite(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen0,axiom,
    type_len0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__0_464,axiom,
    semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ogroup__add_465,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_466,axiom,
    mult_zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_467,axiom,
    comm_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder_468,axiom,
    order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring_469,axiom,
    semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Fields_Oinverse_470,axiom,
    inverse(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oord_471,axiom,
    ord(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ouminus_472,axiom,
    uminus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oabs__if_473,axiom,
    abs_if(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Otimes_474,axiom,
    times(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ominus_475,axiom,
    minus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Power_Opower_476,axiom,
    power(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ozero_477,axiom,
    zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oplus_478,axiom,
    plus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oring_479,axiom,
    ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oone_480,axiom,
    one(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Odvd_481,axiom,
    dvd(finite_1) ).

tff(tcon_Enum_Ofinite__1___HOL_Oequal_482,axiom,
    cl_HOL_Oequal(finite_1) ).

tff(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_483,axiom,
    condit6923001295902523014norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_484,axiom,
    semiri1453513574482234551roduct(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_485,axiom,
    euclid3128863361964157862miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_486,axiom,
    euclid4440199948858584721cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_487,axiom,
    semiri6575147826004484403cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_488,axiom,
    euclid3725896446679973847miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_489,axiom,
    semiri2026040879449505780visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_490,axiom,
    semiri3467727345109120633visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_491,axiom,
    cancel2418104881723323429up_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_492,axiom,
    ring_15535105094025558882visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_493,axiom,
    cancel1802427076303600483id_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_494,axiom,
    comm_s4317794764714335236cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_495,axiom,
    cancel_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_496,axiom,
    ab_semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_497,axiom,
    semiring_1_cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_498,axiom,
    algebraic_semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_499,axiom,
    comm_monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_500,axiom,
    ab_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_501,axiom,
    comm_monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_502,axiom,
    semiring_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_503,axiom,
    comm_semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_504,axiom,
    comm_semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_505,axiom,
    semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_506,axiom,
    semidom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_507,axiom,
    semidom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_508,axiom,
    semiring_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_509,axiom,
    semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Odivision__ring_510,axiom,
    division_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_511,axiom,
    comm_semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Owellorder_512,axiom,
    wellorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__top_513,axiom,
    order_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_514,axiom,
    order_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__group__add_515,axiom,
    ab_group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_516,axiom,
    zero_neq_one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_517,axiom,
    idom_abs_sgn(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Opreorder_518,axiom,
    preorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Olinorder_519,axiom,
    linorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_520,axiom,
    monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_521,axiom,
    idom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__divide_522,axiom,
    idom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_523,axiom,
    comm_ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__add_524,axiom,
    monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_525,axiom,
    finite_finite(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen0_526,axiom,
    type_len0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1_527,axiom,
    semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__0_528,axiom,
    semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ogroup__add_529,axiom,
    group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen_530,axiom,
    type_len(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Omult__zero_531,axiom,
    mult_zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_532,axiom,
    comm_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder_533,axiom,
    order(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Oneg__numeral_534,axiom,
    neg_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring_535,axiom,
    semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Oinverse_536,axiom,
    inverse(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom_537,axiom,
    semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oord_538,axiom,
    ord(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ouminus_539,axiom,
    uminus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1_540,axiom,
    ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Otimes_541,axiom,
    times(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ominus_542,axiom,
    minus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Ofield_543,axiom,
    field(finite_2) ).

tff(tcon_Enum_Ofinite__2___Power_Opower_544,axiom,
    power(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Onumeral_545,axiom,
    numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ozero_546,axiom,
    zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oplus_547,axiom,
    plus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring_548,axiom,
    ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom_549,axiom,
    idom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oone_550,axiom,
    one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Odvd_551,axiom,
    dvd(finite_2) ).

tff(tcon_Enum_Ofinite__2___HOL_Oequal_552,axiom,
    cl_HOL_Oequal(finite_2) ).

tff(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_553,axiom,
    condit6923001295902523014norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_554,axiom,
    semiri1453513574482234551roduct(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_555,axiom,
    euclid3128863361964157862miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_556,axiom,
    euclid4440199948858584721cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_557,axiom,
    semiri6575147826004484403cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_558,axiom,
    euclid3725896446679973847miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_559,axiom,
    semiri2026040879449505780visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_560,axiom,
    semiri3467727345109120633visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_561,axiom,
    cancel2418104881723323429up_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_562,axiom,
    ring_15535105094025558882visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_563,axiom,
    cancel1802427076303600483id_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_564,axiom,
    comm_s4317794764714335236cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_565,axiom,
    cancel_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_566,axiom,
    ab_semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_567,axiom,
    semiring_1_cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_568,axiom,
    algebraic_semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_569,axiom,
    comm_monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_570,axiom,
    ab_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_571,axiom,
    comm_monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_572,axiom,
    semiring_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_573,axiom,
    comm_semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_574,axiom,
    comm_semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_575,axiom,
    semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_576,axiom,
    semidom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_577,axiom,
    semidom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_578,axiom,
    semiring_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_579,axiom,
    semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Odivision__ring_580,axiom,
    division_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_581,axiom,
    comm_semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Owellorder_582,axiom,
    wellorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__top_583,axiom,
    order_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_584,axiom,
    order_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__group__add_585,axiom,
    ab_group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_586,axiom,
    zero_neq_one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_587,axiom,
    idom_abs_sgn(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Opreorder_588,axiom,
    preorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Olinorder_589,axiom,
    linorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_590,axiom,
    monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_591,axiom,
    idom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__divide_592,axiom,
    idom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_593,axiom,
    comm_ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__add_594,axiom,
    monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_595,axiom,
    finite_finite(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen0_596,axiom,
    type_len0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1_597,axiom,
    semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__0_598,axiom,
    semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ogroup__add_599,axiom,
    group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen_600,axiom,
    type_len(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Omult__zero_601,axiom,
    mult_zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_602,axiom,
    comm_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder_603,axiom,
    order(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Oneg__numeral_604,axiom,
    neg_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring_605,axiom,
    semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Oinverse_606,axiom,
    inverse(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom_607,axiom,
    semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oord_608,axiom,
    ord(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ouminus_609,axiom,
    uminus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1_610,axiom,
    ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Otimes_611,axiom,
    times(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ominus_612,axiom,
    minus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Ofield_613,axiom,
    field(finite_3) ).

tff(tcon_Enum_Ofinite__3___Power_Opower_614,axiom,
    power(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Onumeral_615,axiom,
    numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ozero_616,axiom,
    zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oplus_617,axiom,
    plus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring_618,axiom,
    ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom_619,axiom,
    idom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oone_620,axiom,
    one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Odvd_621,axiom,
    dvd(finite_3) ).

tff(tcon_Enum_Ofinite__3___HOL_Oequal_622,axiom,
    cl_HOL_Oequal(finite_3) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_623,axiom,
    ! [A9: $tType] : order_top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_624,axiom,
    ! [A9: $tType] : order_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_625,axiom,
    ! [A9: $tType] : preorder(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_626,axiom,
    ! [A9: $tType] : order(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_627,axiom,
    ! [A9: $tType] : ord(filter(A9)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_628,axiom,
    ! [A9: $tType] :
      ( cl_HOL_Oequal(A9)
     => cl_HOL_Oequal(filter(A9)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_629,axiom,
    ! [A9: $tType] :
      ( comple5582772986160207858norder(A9)
     => condit6923001295902523014norder(option(A9)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_630,axiom,
    ! [A9: $tType] :
      ( wellorder(A9)
     => wellorder(option(A9)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_631,axiom,
    ! [A9: $tType] :
      ( order_top(A9)
     => order_top(option(A9)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_632,axiom,
    ! [A9: $tType] :
      ( order(A9)
     => order_bot(option(A9)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_633,axiom,
    ! [A9: $tType] :
      ( preorder(A9)
     => preorder(option(A9)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_634,axiom,
    ! [A9: $tType] :
      ( linorder(A9)
     => linorder(option(A9)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_635,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(option(A9)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_636,axiom,
    ! [A9: $tType] :
      ( order(A9)
     => order(option(A9)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_637,axiom,
    ! [A9: $tType] :
      ( preorder(A9)
     => ord(option(A9)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_638,axiom,
    ! [A9: $tType] :
      ( heap(A9)
     => heap(option(A9)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_639,axiom,
    ! [A9: $tType] : cl_HOL_Oequal(option(A9)) ).

tff(tcon_Option_Ooption___Nat_Osize_640,axiom,
    ! [A9: $tType] : size(option(A9)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_641,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_642,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_643,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_644,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_645,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_646,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_647,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_648,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_649,axiom,
    least_6119777620449941438nt_lsb(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_650,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_651,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_652,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_653,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_654,axiom,
    generic_set_set_bit(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_655,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_656,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_657,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_658,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_659,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_660,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_661,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_662,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_663,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_664,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_665,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Oring__parity_666,axiom,
    ring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_667,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_668,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_669,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_670,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_671,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_672,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_673,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_674,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_675,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_676,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_677,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_678,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_679,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_680,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_681,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_682,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Otimes_683,axiom,
    times(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ominus_684,axiom,
    minus(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_685,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_686,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_687,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_688,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_689,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_690,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_691,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___HOL_Oequal_692,axiom,
    cl_HOL_Oequal(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_693,axiom,
    size(uint32) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_694,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_695,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_696,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_697,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_698,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_699,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_700,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_701,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Heap_Oheap_702,axiom,
    heap(literal) ).

tff(tcon_String_Oliteral___HOL_Oequal_703,axiom,
    cl_HOL_Oequal(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_704,axiom,
    size(literal) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_705,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_706,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_707,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_708,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_709,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_710,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_711,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_712,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_713,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_714,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_715,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_716,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_717,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_718,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_719,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_720,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_721,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_722,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_723,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_724,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_725,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_726,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_727,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_728,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_729,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_730,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_731,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_732,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_733,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_734,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_735,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_736,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_737,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_738,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_739,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_740,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_741,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_742,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_743,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_744,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_745,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_746,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_747,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_748,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_749,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_750,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_751,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_752,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_753,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_754,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_755,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_756,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_757,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_758,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_759,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_760,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_761,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_762,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_763,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_764,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_765,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_766,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_767,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_768,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_769,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_770,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_771,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_772,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_773,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_774,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_775,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_776,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_777,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_778,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_779,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_780,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_781,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_782,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_783,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_784,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_785,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_786,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_787,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_788,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Otimes_789,axiom,
    times(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_790,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_791,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_792,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_793,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_794,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_795,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_796,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_797,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_798,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_799,axiom,
    dvd(complex) ).

tff(tcon_Complex_Ocomplex___HOL_Oequal_800,axiom,
    cl_HOL_Oequal(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_801,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_802,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_803,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_804,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_805,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_806,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_807,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_808,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_809,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_810,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_811,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_812,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_813,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_814,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_815,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_816,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_817,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_818,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_819,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_820,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_821,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_822,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_823,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_824,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_825,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_826,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_827,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_828,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_829,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_830,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_831,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_832,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_833,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_834,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_835,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_836,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_837,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Otimes_838,axiom,
    times(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_839,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_840,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_841,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_842,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_843,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_844,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_845,axiom,
    dvd(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___HOL_Oequal_846,axiom,
    cl_HOL_Oequal(extended_enat) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_847,axiom,
    ! [A9: $tType] :
      ( preorder(A9)
     => ordere6658533253407199908up_add(multiset(A9)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_848,axiom,
    ! [A9: $tType] : cancel2418104881723323429up_add(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_849,axiom,
    ! [A9: $tType] : cancel1802427076303600483id_add(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_850,axiom,
    ! [A9: $tType] : cancel_semigroup_add(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_851,axiom,
    ! [A9: $tType] : comm_monoid_diff(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_852,axiom,
    ! [A9: $tType] : ab_semigroup_add(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_853,axiom,
    ! [A9: $tType] : comm_monoid_add(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_854,axiom,
    ! [A9: $tType] : semigroup_add(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_855,axiom,
    ! [A9: $tType] :
      ( preorder(A9)
     => preorder(multiset(A9)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_856,axiom,
    ! [A9: $tType] : monoid_add(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_857,axiom,
    ! [A9: $tType] :
      ( preorder(A9)
     => order(multiset(A9)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_858,axiom,
    ! [A9: $tType] :
      ( preorder(A9)
     => ord(multiset(A9)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_859,axiom,
    ! [A9: $tType] : minus(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_860,axiom,
    ! [A9: $tType] : zero(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_861,axiom,
    ! [A9: $tType] : plus(multiset(A9)) ).

tff(tcon_Multiset_Omultiset___HOL_Oequal_862,axiom,
    ! [A9: $tType] :
      ( cl_HOL_Oequal(A9)
     => cl_HOL_Oequal(multiset(A9)) ) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_863,axiom,
    ! [A9: $tType] : size(multiset(A9)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_864,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cancel2418104881723323429up_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_865,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cancel1802427076303600483id_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_866,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_s4317794764714335236cancel(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_867,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cancel_semigroup_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_868,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ab_semigroup_mult(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_869,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_1_cancel(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_870,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_monoid_mult(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_871,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ab_semigroup_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_872,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_monoid_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_873,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_semiring_1(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_874,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_semiring_0(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_875,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semigroup_mult(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_876,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_numeral(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_877,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semigroup_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_878,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_semiring(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_879,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => wellorder(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_880,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ab_group_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_881,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => zero_neq_one(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_882,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => preorder(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_883,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => linorder(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_884,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => monoid_mult(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_885,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_ring_1(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_886,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => monoid_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_887,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => card2(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_888,axiom,
    ! [A9: $tType] :
      ( type_len0(A9)
     => type_len0(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_889,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_1(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_890,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_0(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_891,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => group_add(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen_892,axiom,
    ! [A9: $tType] :
      ( type_len(A9)
     => type_len(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_893,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => mult_zero(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_894,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_ring(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_895,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => order(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_896,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => neg_numeral(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_897,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_898,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ord(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_899,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => uminus(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_900,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ring_1(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Otimes_901,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => times(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ominus_902,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => minus(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_903,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => power(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_904,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => numeral(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_905,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => zero(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_906,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => plus(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_907,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ring(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_908,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => one(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_909,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => dvd(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit0___HOL_Oequal_910,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cl_HOL_Oequal(numeral_bit0(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_911,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cancel2418104881723323429up_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_912,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cancel1802427076303600483id_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_913,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_s4317794764714335236cancel(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_914,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cancel_semigroup_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_915,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ab_semigroup_mult(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_916,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_1_cancel(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_917,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_monoid_mult(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_918,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ab_semigroup_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_919,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_monoid_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_920,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_semiring_1(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_921,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_semiring_0(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_922,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semigroup_mult(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_923,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_numeral(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_924,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semigroup_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_925,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_semiring(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_926,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => wellorder(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_927,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ab_group_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_928,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => zero_neq_one(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_929,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => preorder(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_930,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => linorder(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_931,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => monoid_mult(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_932,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_ring_1(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_933,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => monoid_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_934,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_935,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => card2(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_936,axiom,
    ! [A9: $tType] :
      ( type_len0(A9)
     => type_len0(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_937,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_1(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_938,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring_0(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_939,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => group_add(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_940,axiom,
    ! [A9: $tType] :
      ( type_len0(A9)
     => type_len(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_941,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => mult_zero(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_942,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => comm_ring(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_943,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => order(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_944,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => neg_numeral(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_945,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => semiring(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_946,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ord(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_947,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => uminus(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_948,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ring_1(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Otimes_949,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => times(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ominus_950,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => minus(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_951,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => power(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_952,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => numeral(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_953,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => zero(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_954,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => plus(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_955,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => ring(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_956,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => one(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_957,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => dvd(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Obit1___HOL_Oequal_958,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => cl_HOL_Oequal(numeral_bit1(A9)) ) ).

tff(tcon_Numeral__Type_Onum0___Type__Length_Olen0_959,axiom,
    type_len0(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___HOL_Oequal_960,axiom,
    cl_HOL_Oequal(numeral_num0) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_961,axiom,
    cancel2418104881723323429up_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_962,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_963,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_964,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_965,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_966,axiom,
    ab_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_967,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_968,axiom,
    comm_semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_969,axiom,
    semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_970,axiom,
    semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_971,axiom,
    comm_semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_972,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_973,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_974,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_975,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_976,axiom,
    monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Cardinality_OCARD__1,axiom,
    cARD_1(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_977,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_978,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0_979,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_980,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_981,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_982,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_983,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_984,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_985,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring_986,axiom,
    semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_987,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_988,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Otimes_989,axiom,
    times(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ominus_990,axiom,
    minus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_991,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_992,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_993,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_994,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Oring_995,axiom,
    ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_996,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_997,axiom,
    dvd(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___HOL_Oequal_998,axiom,
    cl_HOL_Oequal(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_999,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( topolo4958980785337419405_space(A9)
        & topolo4958980785337419405_space(A10) )
     => topolo4958980785337419405_space(product_prod(A9,A10)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_1000,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( topological_t2_space(A9)
        & topological_t2_space(A10) )
     => topological_t2_space(product_prod(A9,A10)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1001,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A10) )
     => finite_finite(product_prod(A9,A10)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_1002,axiom,
    ! [A9: $tType,A10: $tType] :
      ( ( heap(A9)
        & heap(A10) )
     => heap(product_prod(A9,A10)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_1003,axiom,
    ! [A9: $tType,A10: $tType] : cl_HOL_Oequal(product_prod(A9,A10)) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_1004,axiom,
    ! [A9: $tType,A10: $tType] : size(product_prod(A9,A10)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_1005,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_1006,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_1007,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_1008,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_1009,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_1010,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_1011,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_1012,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_1013,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_1014,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_1015,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_1016,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_1017,axiom,
    heap(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_1018,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_1019,axiom,
    ! [A9: $tType] : cl_HOL_Oequal(heap_ext(A9)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1020,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1021,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1022,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1023,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1024,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1025,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1026,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1027,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1028,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1029,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1030,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1031,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1032,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1033,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1034,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1035,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1036,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1037,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1038,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1039,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1040,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1041,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1042,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1043,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1044,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1045,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1046,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1047,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1048,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1049,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1050,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1051,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1052,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1053,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1054,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1055,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1056,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1057,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1058,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1059,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1060,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1061,axiom,
    least_6119777620449941438nt_lsb(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1062,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1063,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1064,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1065,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1066,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1067,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1068,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1069,axiom,
    generic_set_set_bit(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1070,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1071,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1072,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1073,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1074,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1075,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1076,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1077,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1078,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1079,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1080,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1081,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1082,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1083,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1084,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1085,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1086,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1087,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1088,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1089,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1090,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1091,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1092,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1093,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1094,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1095,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1096,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1097,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1098,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1099,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1100,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1101,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1102,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1103,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1104,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1105,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1106,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1107,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1108,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1109,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1110,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1111,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1112,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_1113,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_1114,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1115,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1116,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1117,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1118,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1119,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1120,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1121,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1122,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_1123,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_1124,axiom,
    ! [A9: $tType] : cl_HOL_Oequal(heap_Time_Heap(A9)) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1125,axiom,
    ! [A9: $tType] : size(heap_Time_Heap(A9)) ).

tff(tcon_VEBT__Definitions_OVEBT___HOL_Oequal_1126,axiom,
    cl_HOL_Oequal(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1127,axiom,
    size(vEBT_VEBT) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1128,axiom,
    heap(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___HOL_Oequal_1129,axiom,
    cl_HOL_Oequal(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1130,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,fequal(A,X9),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ~ aa(A,$o,fequal(A,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X8: A] : aa(A,$o,P,X8) ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

tff(tfree_1,hypothesis,
    linorder(b) ).

% Conjectures (1)
tff(conj_0,conjecture,
    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(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,xa),x)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,replicate(vEBT_VEBT,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))),divide_divide(nat,na,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,na,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),x))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),xb))),pure_assn(xc = vEBT_Nodei(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,na)),xa,xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,na)))),xc)) ).

%------------------------------------------------------------------------------