%------------------------------------------------------------------------------
% File     : ITP285_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00107_006275
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0094_VEBT_SuccPredImperative_00107_006275 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11526 (4517 unt;1391 typ;   0 def)
%            Number of atoms       : 16743 (8544 equ)
%            Maximal formula atoms :   39 (   1 avg)
%            Number of connectives : 19299 (2055   ~; 325   |;1649   &)
%                                         (2050 <=>;13220  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   34 (   2 avg)
%            Number of FOOLs       : 1498 ( 739 fml; 759 var)
%            Number of X terms     : 1070 (   0  []; 819 ite; 251 let)
%            Number of types       :   18 (  17 usr)
%            Number of type conns  : 1192 ( 980   >; 212   *;   0   +;   0  <<)
%            Number of predicates  :  215 ( 212 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1188 (1188 usr;  92 con; 0-6 aty)
%            Number of variables   : 32369 (29646   !; 714   ?;32369   :)
%                                         (2009  !>;   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 19:47:45.186
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
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_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_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_Heap_Oarray,type,
    array: $tType > $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (1359)
tff(sy_cl_Lattices_Obounded__lattice__top,type,
    bounded_lattice_top: 
      !>[A: $tType] : $o ).

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

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

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

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

tff(sy_cl_Lattices_Osup,type,
    sup: 
      !>[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_Obot,type,
    bot: 
      !>[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_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_Lattices_Olattice,type,
    lattice: 
      !>[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_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_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_Enum_Ofinite__lattice,type,
    finite_lattice: 
      !>[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_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_Complete__Lattices_OInf,type,
    complete_Inf: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Lattices_Osemilattice__sup,type,
    semilattice_sup: 
      !>[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_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_Complete__Partial__Order_Occpo,type,
    comple9053668089753744459l_ccpo: 
      !>[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_Lattices_Obounded__lattice__bot,type,
    bounded_lattice_bot: 
      !>[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_Rings_Oring__1__no__zero__divisors,type,
    ring_15535105094025558882visors: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
    cancel2418104881723323429up_add: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

tff(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
    real_V6157519004096292374lgebra: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Topological__Spaces_Operfect__space,type,
    topolo8386298272705272623_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_Bit__Operations_Oring__bit__operations,type,
    bit_ri3973907225187159222ations: 
      !>[A: $tType] : $o ).

tff(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
    comple6319245703460814977attice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

tff(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
    bounde4346867609351753570nf_top: 
      !>[A: $tType] : $o ).

tff(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
    bounde4967611905675639751up_bot: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
    comple489889107523837845lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
    comple592849572758109894attice: 
      !>[A: $tType] : $o ).

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

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

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

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

tff(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
    condit5016429287641298734tinuum: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
    counta3822494911875563373attice: 
      !>[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__lattice,type,
    condit1219197933456340205attice: 
      !>[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] : fun(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] : fun(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] : ( 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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__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] : fun(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] : fun(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] : fun(A,B) ).

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

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

tff(sy_c_ATP_058Lamp__ay____,type,
    aTP_Lamp_ay: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__az____,type,
    aTP_Lamp_az: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ba____,type,
    aTP_Lamp_ba: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bb____,type,
    aTP_Lamp_bb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bc____,type,
    aTP_Lamp_bc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bd____,type,
    aTP_Lamp_bd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__be____,type,
    aTP_Lamp_be: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bf____,type,
    aTP_Lamp_bf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bg____,type,
    aTP_Lamp_bg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__cd____,type,
    aTP_Lamp_cd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ce____,type,
    aTP_Lamp_ce: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cf____,type,
    aTP_Lamp_cf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cg____,type,
    aTP_Lamp_cg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ch____,type,
    aTP_Lamp_ch: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__cr____,type,
    aTP_Lamp_cr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cs____,type,
    aTP_Lamp_cs: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

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

tff(sy_c_ATP_058Lamp__dv____,type,
    aTP_Lamp_dv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__dw____,type,
    aTP_Lamp_dw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__dx____,type,
    aTP_Lamp_dx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dy____,type,
    aTP_Lamp_dy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dz____,type,
    aTP_Lamp_dz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ea____,type,
    aTP_Lamp_ea: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eb____,type,
    aTP_Lamp_eb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ec____,type,
    aTP_Lamp_ec: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ed____,type,
    aTP_Lamp_ed: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ee____,type,
    aTP_Lamp_ee: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ef____,type,
    aTP_Lamp_ef: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eg____,type,
    aTP_Lamp_eg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ft____,type,
    aTP_Lamp_ft: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fu____,type,
    aTP_Lamp_fu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fv____,type,
    aTP_Lamp_fv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fw____,type,
    aTP_Lamp_fw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fx____,type,
    aTP_Lamp_fx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fy____,type,
    aTP_Lamp_fy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fz____,type,
    aTP_Lamp_fz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ga____,type,
    aTP_Lamp_ga: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gb____,type,
    aTP_Lamp_gb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gc____,type,
    aTP_Lamp_gc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gd____,type,
    aTP_Lamp_gd: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ji____,type,
    aTP_Lamp_ji: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jj____,type,
    aTP_Lamp_jj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jk____,type,
    aTP_Lamp_jk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jl____,type,
    aTP_Lamp_jl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jm____,type,
    aTP_Lamp_jm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jn____,type,
    aTP_Lamp_jn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jo____,type,
    aTP_Lamp_jo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jp____,type,
    aTP_Lamp_jp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jq____,type,
    aTP_Lamp_jq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jr____,type,
    aTP_Lamp_jr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__js____,type,
    aTP_Lamp_js: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jt____,type,
    aTP_Lamp_jt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

tff(sy_c_ATP_058Lamp__jx____,type,
    aTP_Lamp_jx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jy____,type,
    aTP_Lamp_jy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jz____,type,
    aTP_Lamp_jz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ka____,type,
    aTP_Lamp_ka: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kb____,type,
    aTP_Lamp_kb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kc____,type,
    aTP_Lamp_kc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kd____,type,
    aTP_Lamp_kd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ke____,type,
    aTP_Lamp_ke: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kf____,type,
    aTP_Lamp_kf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kg____,type,
    aTP_Lamp_kg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kh____,type,
    aTP_Lamp_kh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ki____,type,
    aTP_Lamp_ki: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kj____,type,
    aTP_Lamp_kj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kk____,type,
    aTP_Lamp_kk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kl____,type,
    aTP_Lamp_kl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__km____,type,
    aTP_Lamp_km: 
      !>[A: $tType,B: $tType] : 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] : ( A > B ) ).

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__kv____,type,
    aTP_Lamp_kv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kw____,type,
    aTP_Lamp_kw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kx____,type,
    aTP_Lamp_kx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ky____,type,
    aTP_Lamp_ky: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kz____,type,
    aTP_Lamp_kz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__la____,type,
    aTP_Lamp_la: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lb____,type,
    aTP_Lamp_lb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lc____,type,
    aTP_Lamp_lc: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lf____,type,
    aTP_Lamp_lf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lg____,type,
    aTP_Lamp_lg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lh____,type,
    aTP_Lamp_lh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__li____,type,
    aTP_Lamp_li: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lj____,type,
    aTP_Lamp_lj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lk____,type,
    aTP_Lamp_lk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ll____,type,
    aTP_Lamp_ll: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lm____,type,
    aTP_Lamp_lm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ln____,type,
    aTP_Lamp_ln: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : ( A > B ) ).

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

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

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

tff(sy_c_ATP_058Lamp__mk____,type,
    aTP_Lamp_mk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ml____,type,
    aTP_Lamp_ml: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mm____,type,
    aTP_Lamp_mm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mn____,type,
    aTP_Lamp_mn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mo____,type,
    aTP_Lamp_mo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mp____,type,
    aTP_Lamp_mp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mq____,type,
    aTP_Lamp_mq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mr____,type,
    aTP_Lamp_mr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ms____,type,
    aTP_Lamp_ms: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mt____,type,
    aTP_Lamp_mt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mu____,type,
    aTP_Lamp_mu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mv____,type,
    aTP_Lamp_mv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mw____,type,
    aTP_Lamp_mw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mx____,type,
    aTP_Lamp_mx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__my____,type,
    aTP_Lamp_my: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mz____,type,
    aTP_Lamp_mz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__na____,type,
    aTP_Lamp_na: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nb____,type,
    aTP_Lamp_nb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nc____,type,
    aTP_Lamp_nc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nd____,type,
    aTP_Lamp_nd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ne____,type,
    aTP_Lamp_ne: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nf____,type,
    aTP_Lamp_nf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ng____,type,
    aTP_Lamp_ng: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nh____,type,
    aTP_Lamp_nh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ni____,type,
    aTP_Lamp_ni: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nj____,type,
    aTP_Lamp_nj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nk____,type,
    aTP_Lamp_nk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nl____,type,
    aTP_Lamp_nl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__nn____,type,
    aTP_Lamp_nn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__no____,type,
    aTP_Lamp_no: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__np____,type,
    aTP_Lamp_np: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nq____,type,
    aTP_Lamp_nq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nr____,type,
    aTP_Lamp_nr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ns____,type,
    aTP_Lamp_ns: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nt____,type,
    aTP_Lamp_nt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nu____,type,
    aTP_Lamp_nu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nv____,type,
    aTP_Lamp_nv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nw____,type,
    aTP_Lamp_nw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nx____,type,
    aTP_Lamp_nx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ny____,type,
    aTP_Lamp_ny: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nz____,type,
    aTP_Lamp_nz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oa____,type,
    aTP_Lamp_oa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ob____,type,
    aTP_Lamp_ob: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oc____,type,
    aTP_Lamp_oc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__od____,type,
    aTP_Lamp_od: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oe____,type,
    aTP_Lamp_oe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__of____,type,
    aTP_Lamp_of: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__og____,type,
    aTP_Lamp_og: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oh____,type,
    aTP_Lamp_oh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oi____,type,
    aTP_Lamp_oi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oj____,type,
    aTP_Lamp_oj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ok____,type,
    aTP_Lamp_ok: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__si____,type,
    aTP_Lamp_si: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sj____,type,
    aTP_Lamp_sj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sk____,type,
    aTP_Lamp_sk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sl____,type,
    aTP_Lamp_sl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sm____,type,
    aTP_Lamp_sm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sn____,type,
    aTP_Lamp_sn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__so____,type,
    aTP_Lamp_so: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sp____,type,
    aTP_Lamp_sp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sq____,type,
    aTP_Lamp_sq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sr____,type,
    aTP_Lamp_sr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ss____,type,
    aTP_Lamp_ss: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__st____,type,
    aTP_Lamp_st: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__su____,type,
    aTP_Lamp_su: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sv____,type,
    aTP_Lamp_sv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sw____,type,
    aTP_Lamp_sw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sx____,type,
    aTP_Lamp_sx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sy____,type,
    aTP_Lamp_sy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sz____,type,
    aTP_Lamp_sz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ta____,type,
    aTP_Lamp_ta: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tb____,type,
    aTP_Lamp_tb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tc____,type,
    aTP_Lamp_tc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__td____,type,
    aTP_Lamp_td: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__te____,type,
    aTP_Lamp_te: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tf____,type,
    aTP_Lamp_tf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__tq____,type,
    aTP_Lamp_tq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tr____,type,
    aTP_Lamp_tr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ts____,type,
    aTP_Lamp_ts: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tt____,type,
    aTP_Lamp_tt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tu____,type,
    aTP_Lamp_tu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tv____,type,
    aTP_Lamp_tv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tw____,type,
    aTP_Lamp_tw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tx____,type,
    aTP_Lamp_tx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ty____,type,
    aTP_Lamp_ty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tz____,type,
    aTP_Lamp_tz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uc____,type,
    aTP_Lamp_uc: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uf____,type,
    aTP_Lamp_uf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ug____,type,
    aTP_Lamp_ug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ul____,type,
    aTP_Lamp_ul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__us____,type,
    aTP_Lamp_us: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ut____,type,
    aTP_Lamp_ut: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xn____,type,
    aTP_Lamp_xn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__xq____,type,
    aTP_Lamp_xq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xr____,type,
    aTP_Lamp_xr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xs____,type,
    aTP_Lamp_xs: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yd____,type,
    aTP_Lamp_yd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ye____,type,
    aTP_Lamp_ye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yf____,type,
    aTP_Lamp_yf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Array__Time_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_Omap__entry,type,
    array_map_entry: 
      !>[A: $tType] : ( ( nat * fun(A,A) * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onew,type,
    array_new: 
      !>[A: $tType] : ( ( nat * A ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onth,type,
    array_nth: 
      !>[A: $tType] : ( ( array(A) * 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_Oswap,type,
    array_swap: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Oupd,type,
    array_upd: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Assertions_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_Oprecise,type,
    precise: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,assn)) > $o ) ).

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_BNF__Greatest__Fixpoint_OSucc,type,
    bNF_Greatest_Succ: 
      !>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).

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

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

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

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

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

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

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

tff(sy_c_Bit__Operations_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] : ( ( nat * A ) > A ) ).

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

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

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

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

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
    code_T4080844693773952564amples: 
      !>[A: $tType] : ( ( fun(code_integer,product_prod(A,A)) * A * code_integer ) > list(A) ) ).

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

tff(sy_c_Code__Target__Word__Base_Oset__bits__aux,type,
    code_T2661198915054445665ts_aux: 
      !>[A: $tType] : ( ( fun(nat,$o) * nat * word(A) ) > word(A) ) ).

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
    complete_Sup_Sup: 
      !>[A: $tType] : fun(set(A),A) ).

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

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

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

tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

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_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

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

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

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

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

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

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

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : ( 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] : ( A > A ) ).

tff(sy_c_Groups_Ozero__class_Ozero,type,
    zero_zero: 
      !>[A: $tType] : A ).

tff(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
    groups3894954378712506084id_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A * fun(B,A) * set(B) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : fun(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(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A ) > fun(list(B),A) ) ).

tff(sy_c_Groups__List_Omonoid__add_Osum__list,type,
    groups4543113879258116180m_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * list(A) ) > A ) ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : ( list(A) > A ) ).

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

tff(sy_c_Hash__Instances_Ohash__code__option,type,
    hash_h1887023736457453652option: 
      !>[A: $tType] : ( ( fun(A,uint32) * option(A) ) > uint32 ) ).

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_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Onat,type,
    nat2: int > nat ).

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

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

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

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

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

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

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

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

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

tff(sy_c_List_Obutlast,type,
    butlast: 
      !>[A: $tType] : ( 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_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

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

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),list(A)) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(A) ) ).

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

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

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olinorder_Oinsort__insert__key,type,
    insort_insert_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) ) > fun(B,fun(list(B),list(B))) ) ).

tff(sy_c_List_Olinorder_Osorted__list__of__set,type,
    sorted_list_of_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > list(A) ) ).

tff(sy_c_List_Olinorder_Ostable__sort__key,type,
    stable_sort_key: 
      !>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).

tff(sy_c_List_Olinorder__class_Oinsort__key,type,
    linorder_insort_key: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(B,fun(list(B),list(B))) ) ).

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

tff(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
    linord144544945434240204of_set: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),list(B)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : fun(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) * list(A) ) > list(Aa) ) ).

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

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).

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

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

tff(sy_c_List_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_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_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

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

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

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : fun(list(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_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

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

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

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > 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__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

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

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

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

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

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Misc_Obijective,type,
    bijective: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

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

tff(sy_c_Misc_Omerge,type,
    merge: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Omerge__list,type,
    merge_list: 
      !>[A: $tType] : ( ( list(list(A)) * list(list(A)) ) > list(A) ) ).

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

tff(sy_c_Misc_Omergesort__remdups,type,
    mergesort_remdups: 
      !>[A: $tType] : ( list(A) > list(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_Oremove__rev,type,
    remove_rev: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

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

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

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

tff(sy_c_Multiset_Osubseteq__mset,type,
    subseteq_mset: 
      !>[A: $tType] : ( ( multiset(A) * multiset(A) ) > $o ) ).

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

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

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

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

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

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: fun(list(nat),nat) ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),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_Osqrt,type,
    sqrt: 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__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

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

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

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_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_Obit0_OAbs__bit0,type,
    numeral_Abs_bit0: 
      !>[A: $tType] : fun(int,numeral_bit0(A)) ).

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

tff(sy_c_Numeral__Type_Obit1_OAbs__bit1,type,
    numeral_Abs_bit1: 
      !>[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_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

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

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

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

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

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

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] : fun(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(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_Quicksort_Olinorder__class_Oquicksort,type,
    linorder_quicksort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
    linord6200660962353139674rt_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

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

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

tff(sy_c_Refine__Imp__Hol_Orefines,type,
    refine_Imp_refines: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_Time_Heap(A) ) > $o ) ).

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

tff(sy_c_Relation_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

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

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

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

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

tff(sy_c_Set_OPow,type,
    pow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

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

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

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

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

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

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(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_Time__Reasoning_OTBOUND,type,
    time_TBOUND: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * nat ) > $o ) ).

tff(sy_c_Time__Reasoning_Ofails,type,
    time_fails: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(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_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_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_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi,type,
    vEBT_case_VEBTi: 
      !>[A: $tType] : ( ( fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))) * fun($o,fun($o,A)) * vEBT_VEBTi ) > A ) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun(vEBT_VEBT,fun(vEBT_VEBTi,assn)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: ( vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_VEBT__List__Assn_Olist__assn__rel,type,
    vEBT_L4249061453398456502sn_rel: 
      !>[A: $tType,C: $tType] : fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),$o)) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

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

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

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

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

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

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

tff(sy_c_Word_Ocast,type,
    cast: 
      !>[A: $tType,B: $tType] : ( word(A) > word(B) ) ).

tff(sy_c_Word_Oeven__word,type,
    even_word: 
      !>[A: $tType] : fun(word(A),$o) ).

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

tff(sy_c_Word_Oof__nat,type,
    of_nat: 
      !>[A: $tType] : ( nat > word(A) ) ).

tff(sy_c_Word_Orevcast,type,
    revcast: 
      !>[A: $tType,B: $tType] : fun(word(A),word(B)) ).

tff(sy_c_Word_Oring__1__class_Osigned,type,
    ring_1_signed: 
      !>[B: $tType,A: $tType] : ( word(B) > A ) ).

tff(sy_c_Word_Osemiring__1__class_Ounsigned,type,
    semiring_1_unsigned: 
      !>[B: $tType,A: $tType] : fun(word(B),A) ).

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

tff(sy_c_Word_Osigned__drop__bit,type,
    signed_drop_bit: 
      !>[A: $tType] : ( ( nat * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oslice,type,
    slice2: 
      !>[A: $tType,B: $tType] : ( nat > fun(word(A),word(B)) ) ).

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

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

tff(sy_c_Word_Othe__nat,type,
    the_nat: 
      !>[A: $tType] : ( word(A) > nat ) ).

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

tff(sy_c_Word_Oudvd,type,
    udvd: 
      !>[A: $tType] : ( ( word(A) * word(A) ) > $o ) ).

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__rcat,type,
    word_rcat: 
      !>[A: $tType,B: $tType] : fun(list(word(A)),word(B)) ).

tff(sy_c_Word_Oword__reverse,type,
    word_reverse: 
      !>[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: $tType] : fun(word(A),fun(word(A),$o)) ).

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

tff(sy_c_fNot,type,
    fNot: fun($o,$o) ).

tff(sy_c_fconj,type,
    fconj: fun($o,fun($o,$o)) ).

tff(sy_c_fdisj,type,
    fdisj: fun($o,fun($o,$o)) ).

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

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

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_v____,type,
    v: product_prod(nat,nat) ).

tff(sy_v_vg____,type,
    vg: list(vEBT_VEBT) ).

tff(sy_v_vh____,type,
    vh: vEBT_VEBT ).

tff(sy_v_vi____,type,
    vi: nat ).

% Relevant facts (8931)
tff(fact_0_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_1_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_2_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_3_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),S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList2,S) ) ).

% deg_SUcn_Node
tff(fact_4__C5_C,axiom,
    vEBT_invar_vebt(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),na) ).

% "5"
tff(fact_5_norm__pre__pure__iff,axiom,
    ! [A: $tType,P: assn,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F,Q)
    <=> ( (B2)
       => hoare_hoare_triple(A,P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_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_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_8_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_9_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_10_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_11_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_12_norm__pre__pure__rule1,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B2)
       => hoare_hoare_triple(A,P,F,Q) )
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_13_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( pure_assn((P)) = pure_assn((Q)) )
    <=> ( (P)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_14_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_15_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_aa(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule
tff(fact_16_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_ab(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule_left
tff(fact_17_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_18_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_19_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_20_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_21_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_22_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_23_is__hoare__triple,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% is_hoare_triple
tff(fact_24_if__rule,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap(A),Q: fun(A,assn),G: heap_Time_Heap(A)] :
      ( ( (B2)
       => hoare_hoare_triple(A,P,F,Q) )
     => ( ( ~ (B2)
         => hoare_hoare_triple(A,P,G,Q) )
       => hoare_hoare_triple(A,P,
            $ite((B2),F,G),
            Q) ) ) ).

% if_rule
tff(fact_25_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_26_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_27_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_28_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_29_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_30_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ac(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_31_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_32_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_33_option_Oinject,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X2) = aa(A,option(A),some(A),Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_34_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_35_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_36_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_37_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_38_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_39_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_40_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(Ta,D)
     => vEBT_invar_vebt(Ta,D) ) ).

% valid_eq2
tff(fact_41_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(Ta,D)
    <=> vEBT_invar_vebt(Ta,D) ) ).

% valid_eq
tff(fact_42_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,A2),aa(fun(A,$o),set(A),collect(A),P))
    <=> aa(A,$o,P,A2) ) ).

% mem_Collect_eq
tff(fact_43_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3)) = A3 ).

% Collect_mem_eq
tff(fact_44_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
        <=> aa(A,$o,Q,X) )
     => ( aa(fun(A,$o),set(A),collect(A),P) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_45_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
      ( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
     => ( F = G ) ) ).

% HOL.ext
tff(fact_46_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( vEBT_invar_vebt(Ta,D)
     => vEBT_VEBT_valid(Ta,D) ) ).

% valid_eq1
tff(fact_47_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: A] :
          ( ( zero_zero(A) = Xa )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_48_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_49_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_50_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_51_Suc__inject,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xa) = aa(nat,nat,suc,Y) )
     => ( Xa = Y ) ) ).

% Suc_inject
tff(fact_52_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_53_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_54_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_55_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_56_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_57_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_58_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_59_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_60_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_61_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Nb: nat] :
      ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X),zero_zero(nat))
     => ( ! [Y3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y3))
       => ( ! [X: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X),Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X)),aa(nat,nat,suc,Y3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Nb) ) ) ) ).

% diff_induct
tff(fact_62_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_63_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_64_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_65_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_66_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% not0_implies_Suc
tff(fact_67_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(Ta),Xa,Sx) ) ) ).

% succ_correct
tff(fact_68_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_69_precise__extr__pure_I2_J,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),P: $o] :
      ( precise(A,B,aa($o,fun(A,fun(B,assn)),aTP_Lamp_ae(fun(A,fun(B,assn)),fun($o,fun(A,fun(B,assn))),R),(P)))
    <=> ( (P)
       => precise(A,B,R) ) ) ).

% precise_extr_pure(2)
tff(fact_70_precise__extr__pure_I1_J,axiom,
    ! [B: $tType,A: $tType,P: $o,R: fun(A,fun(B,assn))] :
      ( precise(A,B,aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_af($o,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),(P)),R))
    <=> ( (P)
       => precise(A,B,R) ) ) ).

% precise_extr_pure(1)
tff(fact_71_list__decode_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero(nat) )
     => ~ ! [N: nat] : Xa != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_72_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,
    ! [Xa: nat] :
      ( ( Xa != zero_zero(nat) )
     => ( ( Xa != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : Xa != 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_73_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_1)
     => ( ! [X: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X)
           => ? [Y4: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N)),Y4)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N),X),Y4) ) )
       => ? [F2: fun(nat,A)] :
          ! [N2: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N2),aa(nat,A,F2,N2))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N2),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))) ) ) ) ).

% dependent_nat_choice
tff(fact_74_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_1: nat] : aa(nat,$o,P,X_1)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_75_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% deg_not_0
tff(fact_76_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_77_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_78_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_79_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_80_Suc__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% Suc_less_eq
tff(fact_81_Suc__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb)) ) ).

% Suc_mono
tff(fact_82_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_83_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_84_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% neq0_conv
tff(fact_85_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_86_zero__less__Suc,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Nb)) ).

% zero_less_Suc
tff(fact_87_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_88_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_89_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% mult_less_cancel2
tff(fact_90_nat__0__less__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_91_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V: fun(A,nat),Xa: A] :
      ( ! [X: A] :
          ( ~ aa(A,$o,P,X)
         => ? [Y4: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X))
              & ~ aa(A,$o,P,Y4) ) )
     => aa(A,$o,P,Xa) ) ).

% infinite_descent_measure
tff(fact_92_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,Y4)),aa(A,B,F,X))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_93_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_94_infinite__descent,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
              & ~ aa(nat,$o,P,M3) ) )
     => aa(nat,$o,P,Nb) ) ).

% infinite_descent
tff(fact_95_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(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_96_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_irrefl_nat
tff(fact_97_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,Y4)),aa(A,B,F,X))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_98_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_99_less__not__refl2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
     => ( M != Nb ) ) ).

% less_not_refl2
tff(fact_100_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_not_refl
tff(fact_101_nat__neq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M != Nb )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M) ) ) ).

% nat_neq_iff
tff(fact_102_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_103_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F,N)),aa(nat,A,F,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F,Nb)),aa(nat,A,F,N3)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_104_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,M: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F,N)),aa(nat,A,F,aa(nat,nat,suc,N)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F,Nb)),aa(nat,A,F,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_105_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xa) != aa(A,nat,size_size(A),Y) )
         => ( Xa != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_106_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,X222: $o] : Y != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_107_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X22: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_108_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_109_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Nb)
         => ( Nb != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_110_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Nb),zero_zero(A)) ) ).

% not_less_zero
tff(fact_111_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ( Nb != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb) ) ) ).

% gr_zeroI
tff(fact_112_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,M))
      <=> ( Nb = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_113_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_114_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_115_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_116_Suc__less__SucD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% Suc_less_SucD
tff(fact_117_less__antisym,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,M))
       => ( M = Nb ) ) ) ).

% less_antisym
tff(fact_118_Suc__less__eq2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),M)
    <=> ? [M4: nat] :
          ( ( M = aa(nat,nat,suc,M4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M4) ) ) ).

% Suc_less_eq2
tff(fact_119_Nat_OAll__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,Nb)
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
           => aa(nat,$o,P,I3) ) ) ) ).

% Nat.All_less_Suc
tff(fact_120_not__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_121_less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
        | ( M = Nb ) ) ) ).

% less_Suc_eq
tff(fact_122_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,Nb)
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
            & aa(nat,$o,P,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_123_less__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Nb)) ) ).

% less_SucI
tff(fact_124_less__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
       => ( M = Nb ) ) ) ).

% less_SucE
tff(fact_125_Suc__lessI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => ( ( aa(nat,nat,suc,M) != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Nb) ) ) ).

% Suc_lessI
tff(fact_126_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_127_Suc__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% Suc_lessD
tff(fact_128_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_129_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,$o),Xa: A] :
      ( ! [X: A] :
          ( ( aa(A,nat,V,X) = zero_zero(nat) )
         => aa(A,$o,P,X) )
     => ( ! [X: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X))
           => ( ~ aa(A,$o,P,X)
             => ? [Y4: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X))
                  & ~ aa(A,$o,P,Y4) ) ) )
       => aa(A,$o,P,Xa) ) ) ).

% infinite_descent0_measure
tff(fact_130_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
                  & ~ aa(nat,$o,P,M3) ) ) )
       => aa(nat,$o,P,Nb) ) ) ).

% infinite_descent0
tff(fact_131_gr__implies__not0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => ( Nb != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_132_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_133_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% not_less0
tff(fact_134_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_135_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% gr0I
tff(fact_136_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_137_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_138_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_139_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(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_140_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_141_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(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_142_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_143_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_144_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(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_145_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_146_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_147_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_148_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_149_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_150_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_151_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_152_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(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_153_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_154_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(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_155_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_156_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(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_157_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_158_option_Osize__neq,axiom,
    ! [A: $tType,Xa: option(A)] : aa(option(A),nat,size_size(option(A)),Xa) != zero_zero(nat) ).

% option.size_neq
tff(fact_159_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Nb))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_160_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% gr0_implies_Suc
tff(fact_161_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
           => aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_162_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ? [M5: nat] : Nb = aa(nat,nat,suc,M5) ) ).

% gr0_conv_Suc
tff(fact_163_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
            & aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_164_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_165_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$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_166_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$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_167_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_168_n__less__n__mult__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$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_169_n__less__m__mult__n,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$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_170_one__less__mult,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$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_171_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta),Xa,Sx) ) ) ).

% succ_corr
tff(fact_172_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_173_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),aa(A,option(A),some(A),Xa)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% less_option_Some
tff(fact_174_greater__shift,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
    <=> vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Xa),aa(nat,option(nat),some(nat),Y)) ) ).

% greater_shift
tff(fact_175_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Nb),Nb) ) ).

% buildup_gives_valid
tff(fact_176_T__vebt__buildupi__gq__0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),vEBT_V441764108873111860ildupi(Nb)) ).

% T_vebt_buildupi_gq_0
tff(fact_177_less__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
    <=> vEBT_VEBT_less(aa(nat,option(nat),some(nat),Xa),aa(nat,option(nat),some(nat),Y)) ) ).

% less_shift
tff(fact_178_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xa: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Xa))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Xa)
           => aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_179_forall__finite_I2_J,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,zero_zero(nat)))
         => aa(nat,$o,P,I3) )
    <=> aa(nat,$o,P,zero_zero(nat)) ) ).

% forall_finite(2)
tff(fact_180_forall__finite_I3_J,axiom,
    ! [Xa: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,aa(nat,nat,suc,Xa)))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Xa))
           => aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% forall_finite(3)
tff(fact_181_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% mult_less_iff1
tff(fact_182_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Ta),Xa)
      <=> aa(set(nat),$o,member(nat,Xa),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_183_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_184_vebt__memberi_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_V854960066525838166emberi(Ta,Tib,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_ag(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Ta),Tib),Xa)) ).

% vebt_memberi'_rf_abstr
tff(fact_185_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_186_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_187_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_188_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_189_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X22))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_190_forall__finite_I1_J,axiom,
    ! [P: fun(nat,$o),I4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),zero_zero(nat))
     => aa(nat,$o,P,I4) ) ).

% forall_finite(1)
tff(fact_191_vebt__member_Osimps_I4_J,axiom,
    ! [Va2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xa) ).

% vebt_member.simps(4)
tff(fact_192_word__size__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,size_size(word(A)),W)) ) ).

% word_size_gt_0
tff(fact_193_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta),Xa,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z2) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Z2) ) ) ) ).

% succ_member
tff(fact_194_vebt__member_Osimps_I3_J,axiom,
    ! [Va2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),zero_zero(nat),Uy,Uz)),Xa) ).

% vebt_member.simps(3)
tff(fact_195_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xa: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xa) ).

% buildup_nothing_in_leaf
tff(fact_196_mul__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),Xa)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% mul_shift
tff(fact_197_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M6: multiset(A)] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M6) = zero_zero(nat) )
    <=> ( M6 = zero_zero(multiset(A)) ) ) ).

% size_eq_0_iff_empty
tff(fact_198_size__empty,axiom,
    ! [A: $tType] : aa(multiset(A),nat,size_size(multiset(A)),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_199_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_200_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_201_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xa: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb),Xa) ).

% buildup_nothing_in_min_max
tff(fact_202_max__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xa)
    <=> ( aa(set(nat),$o,member(nat,Xa),Xs)
        & ! [X3: nat] :
            ( aa(set(nat),$o,member(nat,X3),Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Xa) ) ) ) ).

% max_in_set_def
tff(fact_203_min__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xa)
    <=> ( aa(set(nat),$o,member(nat,Xa),Xs)
        & ! [X3: nat] :
            ( aa(set(nat),$o,member(nat,X3),Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X3) ) ) ) ).

% min_in_set_def
tff(fact_204_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Tree,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
       => ( vEBT_V5719532721284313246member(Tree,Xa)
          | vEBT_VEBT_membermima(Tree,Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_205_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(Ta),Xa)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_206_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),zero_zero(A))
        <=> ( Nb = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_207_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),Xa)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% less_eq_option_Some
tff(fact_208_Suc__le__mono,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ).

% Suc_le_mono
tff(fact_209_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_210_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_211_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xa) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta),Xa) ) ) ).

% maxt_corr
tff(fact_212_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta),Xa)
       => ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xa) ) ) ) ).

% maxt_sound
tff(fact_213_one__le__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) ) ) ).

% one_le_mult_iff
tff(fact_214_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_215_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% mult_le_cancel2
tff(fact_216_lesseq__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y)
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),Xa),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_217_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xa,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z2),Xa) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Z2),Y) ) ) ) ).

% pred_member
tff(fact_218_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_219_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => ? [X: nat] :
            ( aa(nat,$o,P,X)
            & ! [Y4: nat] :
                ( aa(nat,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_220_nat__le__linear,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ).

% nat_le_linear
tff(fact_221_le__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
       => ( M = Nb ) ) ) ).

% le_antisym
tff(fact_222_eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = Nb )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% eq_imp_le
tff(fact_223_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_224_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_225_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N)),aa(nat,A,F,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,Nb)),aa(nat,A,F,N3)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_226_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,aa(nat,nat,suc,N))),aa(nat,A,F,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N3)),aa(nat,A,F,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_227_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ).

% zero_le
tff(fact_228_transitive__stepwise__le,axiom,
    ! [M: nat,Nb: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X),X)
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y3),Z3)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X),Z3) ) )
         => ( ! [N: nat] : aa(nat,$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_229_nat__induct__at__least,axiom,
    ! [M: nat,Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,P,M)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$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_230_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(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_231_not__less__eq__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M) ) ).

% not_less_eq_eq
tff(fact_232_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_233_le__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        | ( M = aa(nat,nat,suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_234_Suc__le__D,axiom,
    ! [Nb: nat,M7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M7)
     => ? [M2: nat] : M7 = aa(nat,nat,suc,M2) ) ).

% Suc_le_D
tff(fact_235_le__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb)) ) ).

% le_SucI
tff(fact_236_le__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ( M = aa(nat,nat,suc,Nb) ) ) ) ).

% le_SucE
tff(fact_237_Suc__leD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% Suc_leD
tff(fact_238_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),zero_zero(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_239_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_240_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_241_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% less_eq_nat.simps(1)
tff(fact_242_less__mono__imp__le__mono,axiom,
    ! [F: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F,I2)),aa(nat,nat,F,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F,I)),aa(nat,nat,F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_243_le__neq__implies__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( ( M != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% le_neq_implies_less
tff(fact_244_less__or__eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
        | ( M = Nb ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_or_eq_imp_le
tff(fact_245_le__eq__less__or__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
        | ( M = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_246_less__imp__le__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_imp_le_nat
tff(fact_247_nat__less__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & ( M != Nb ) ) ) ).

% nat_less_le
tff(fact_248_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_249_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_250_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_251_le__square,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_252_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_cube
tff(fact_253_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_254_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_255_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_256_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_257_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_258_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_259_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_260_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_261_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_262_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_263_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_264_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_265_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_266_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_267_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% split_mult_pos_le
tff(fact_268_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)) ) ).

% zero_le_square
tff(fact_269_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ) ).

% mult_mono'
tff(fact_270_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ) ).

% mult_mono
tff(fact_271_Suc__leI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb) ) ).

% Suc_leI
tff(fact_272_Suc__le__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% Suc_le_eq
tff(fact_273_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$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_274_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$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_275_Suc__le__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% Suc_le_lessD
tff(fact_276_le__less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,M))
      <=> ( Nb = M ) ) ) ).

% le_less_Suc_eq
tff(fact_277_less__Suc__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_Suc_eq_le
tff(fact_278_less__eq__Suc__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M) ) ).

% less_eq_Suc_le
tff(fact_279_le__imp__less__Suc,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_280_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,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K2)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_281_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_282_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_283_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(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_284_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(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_285_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_286_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_287_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_288_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_289_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_290_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(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_291_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_292_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_293_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(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_294_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_295_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_296_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_297_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_298_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_299_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,aa(nat,fun(nat,$o),ord_less(nat),K2),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(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_300_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_301_size__0__same_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Va2: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( W = Va2 ) ) ) ).

% size_0_same'
tff(fact_302_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_303_size__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Va2: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( Va2 = W ) ) ) ).

% size_0_eq
tff(fact_304_nonempty__has__size,axiom,
    ! [A: $tType,S3: multiset(A)] :
      ( ( S3 != zero_zero(multiset(A)) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S3)) ) ).

% nonempty_has_size
tff(fact_305_vebt__maxti__h,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_maxti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ah(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib)) ).

% vebt_maxti_h
tff(fact_306_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xa)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_307_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_308_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,Xa,Y)
    <=> ( aa(set(nat),$o,member(nat,Y),Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
        & ! [X3: nat] :
            ( aa(set(nat),$o,member(nat,X3),Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),X3)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X3) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_309_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_310_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = aa(nat,nat,suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_311_nat__compl__induct_H,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$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_312_nat__compl__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$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_313_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(Ta),Xa)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mini),Xa) ) ) ) ).

% mint_corr_help
tff(fact_314_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ aa(set(A),$o,member(A,C2),bot_bot(set(A))) ).

% empty_iff
tff(fact_315_all__not__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : ~ aa(set(A),$o,member(A,X3),A3)
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_316_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_317_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A3) ).

% empty_subsetI
tff(fact_318_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A)))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_319_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P) )
    <=> ! [X3: A] : ~ aa(A,$o,P,X3) ) ).

% empty_Collect_eq
tff(fact_320_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ! [X3: A] : ~ aa(A,$o,P,X3) ) ).

% Collect_empty_eq
tff(fact_321_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_322_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Xa) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta),Xa) ) ) ).

% mint_corr
tff(fact_323_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta),Xa)
       => ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Xa) ) ) ) ).

% mint_sound
tff(fact_324_not__None__eq,axiom,
    ! [A: $tType,Xa: option(A)] :
      ( ( Xa != none(A) )
    <=> ? [Y5: A] : Xa = aa(A,option(A),some(A),Y5) ) ).

% not_None_eq
tff(fact_325_not__Some__eq,axiom,
    ! [A: $tType,Xa: option(A)] :
      ( ! [Y5: A] : Xa != aa(A,option(A),some(A),Y5)
    <=> ( Xa = none(A) ) ) ).

% not_Some_eq
tff(fact_326_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),none(A)),Xa) ) ).

% less_eq_option_None_code
tff(fact_327_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] : ~ aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),Xa),none(A)) ) ).

% less_option_None
tff(fact_328_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : ~ aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),Xa)),none(A)) ) ).

% less_eq_option_Some_None
tff(fact_329_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),aa(A,option(A),some(A),Xa)) ) ).

% less_option_None_Some_code
tff(fact_330_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),Xa),none(A))
         => ( Xa = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_331_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),none(A)),Xa) ) ).

% less_eq_option_None
tff(fact_332_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X2: A] : none(A) != aa(A,option(A),some(A),X2) ).

% option.distinct(1)
tff(fact_333_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_334_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X23: A] : Y != aa(A,option(A),some(A),X23) ) ).

% option.exhaust
tff(fact_335_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ? [X_12: option(A)] : aa(option(A),$o,P,X_12)
    <=> ( aa(option(A),$o,P,none(A))
        | ? [X3: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X3)) ) ) ).

% split_option_ex
tff(fact_336_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ! [X_12: option(A)] : aa(option(A),$o,P,X_12)
    <=> ( aa(option(A),$o,P,none(A))
        & ! [X3: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X3)) ) ) ).

% split_option_all
tff(fact_337_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,Xa: option(A),P: fun(option(A),fun(option(B),$o)),Y: option(B)] :
      ( ( ( Xa = none(A) )
       => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xa),Y) )
     => ( ( ( Y = none(B) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xa),Y) )
       => ( ! [A4: A,B3: B] :
              ( ( Xa = aa(A,option(A),some(A),A4) )
             => ( ( Y = aa(B,option(B),some(B),B3) )
               => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xa),Y) ) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_338_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_339_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),Xa)
         => ? [Z3: A] : Xa = aa(A,option(A),some(A),Z3) ) ) ).

% less_option_None_is_Some
tff(fact_340_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),aa(A,option(A),some(A),Xa)) ) ).

% less_option_None_Some
tff(fact_341_vebt__succ_Osimps_I4_J,axiom,
    ! [Va2: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),zero_zero(nat),Vc,Vd),Ve) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_342_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => ( ( C2 = D )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),D) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_343_memb__imp__not__empty,axiom,
    ! [A: $tType,Xa: A,S3: set(A)] :
      ( aa(set(A),$o,member(A,Xa),S3)
     => ( S3 != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_344_set__notEmptyE,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( S3 != bot_bot(set(A)) )
     => ~ ! [X: A] : ~ aa(set(A),$o,member(A,X),S3) ) ).

% set_notEmptyE
tff(fact_345_ex__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ? [X3: A] : aa(set(A),$o,member(A,X3),A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_346_equals0I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [Y3: A] : ~ aa(set(A),$o,member(A,Y3),A3)
     => ( A3 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_347_equals0D,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ( A3 = bot_bot(set(A)) )
     => ~ aa(set(A),$o,member(A,A2),A3) ) ).

% equals0D
tff(fact_348_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ aa(set(A),$o,member(A,A2),bot_bot(set(A))) ).

% emptyE
tff(fact_349_not__psubset__empty,axiom,
    ! [A: $tType,A3: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_350_vebt__succ_Osimps_I5_J,axiom,
    ! [Va2: product_prod(nat,nat),Vga: list(vEBT_VEBT),Vha: vEBT_VEBT,Via: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vga,Vha),Via) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_351_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_352_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj(A,$o)) ).

% Set.empty_def
tff(fact_353_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),M)),Xa)
         => ~ ! [M8: A] :
                ( ( Xa = aa(A,option(A),some(A),M8) )
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),M8) ) ) ) ).

% le_some_optE
tff(fact_354_exists__leI,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ( ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),Nb)
           => ~ aa(nat,$o,P,N4) )
       => aa(nat,$o,P,Nb) )
     => ? [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),Nb)
          & aa(nat,$o,P,N5) ) ) ).

% exists_leI
tff(fact_355_vebt__minti__h,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_minti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ak(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib)) ).

% vebt_minti_h
tff(fact_356_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_al(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xa)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_357_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(Ta),Xa,Sx) ) ) ).

% pred_correct
tff(fact_358_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Px: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xa,Px) ) ) ).

% pred_corr
tff(fact_359_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_360_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_am(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_361_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,Xa,Y)
    <=> ( aa(set(nat),$o,member(nat,Y),Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
        & ! [X3: nat] :
            ( aa(set(nat),$o,member(nat,X3),Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),Xa)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_362_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw: fun(A,fun(A,A)),Va2: 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),Va2)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_363_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,A)),Xaa: option(A),Xb: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Xa),Xaa),Xb) = Y )
     => ( ( ( Xaa = none(A) )
         => ( Y != none(A) ) )
       => ( ( ? [V2: A] : Xaa = aa(A,option(A),some(A),V2)
           => ( ( Xb = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A4: A] :
                ( ( Xaa = aa(A,option(A),some(A),A4) )
               => ! [B3: A] :
                    ( ( Xb = aa(A,option(A),some(A),B3) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xa,A4),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_364_obtain__set__pred,axiom,
    ! [Z: nat,Xa: nat,A3: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z),Xa)
     => ( vEBT_VEBT_min_in_set(A3,Z)
       => ( aa(set(nat),$o,finite_finite2(nat),A3)
         => ? [X_13: nat] : vEBT_is_pred_in_set(A3,Xa,X_13) ) ) ) ).

% obtain_set_pred
tff(fact_365_obtain__set__succ,axiom,
    ! [Xa: nat,Z: nat,A3: set(nat),B4: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( vEBT_VEBT_max_in_set(A3,Z)
       => ( aa(set(nat),$o,finite_finite2(nat),B4)
         => ( ( A3 = B4 )
           => ? [X_13: nat] : vEBT_is_succ_in_set(A3,Xa,X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_366_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,finite_finite2(nat),vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_367_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_succ_in_set(Xs,A2,X_13)
     => ( aa(set(nat),$o,finite_finite2(nat),Xs)
       => ~ ? [X4: nat] :
              ( aa(set(nat),$o,member(nat,X4),Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),X4) ) ) ) ).

% succ_none_empty
tff(fact_368_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_pred_in_set(Xs,A2,X_13)
     => ( aa(set(nat),$o,finite_finite2(nat),Xs)
       => ~ ? [X4: nat] :
              ( aa(set(nat),$o,member(nat,X4),Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),A2) ) ) ) ).

% pred_none_empty
tff(fact_369_subsetI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(set(A),$o,member(A,X),B4) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ).

% subsetI
tff(fact_370_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
       => ( A3 = B4 ) ) ) ).

% subset_antisym
tff(fact_371_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,Va2: option(product_prod(A,B))] :
      ( ! [X3: A,Y5: B] : Va2 != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y5))
    <=> ( Va2 = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_372_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( ( A3 != B4 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4) ) ) ).

% psubsetI
tff(fact_373_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3) ) ) ).

% psubsetE
tff(fact_374_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
        & ( A3 != B4 ) ) ) ).

% psubset_eq
tff(fact_375_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ).

% psubset_imp_subset
tff(fact_376_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% psubset_subset_trans
tff(fact_377_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3) ) ) ).

% subset_not_subset_eq
tff(fact_378_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B4),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% subset_psubset_trans
tff(fact_379_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
        | ( A3 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_380_in__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,member(A,Xa),A3)
       => aa(set(A),$o,member(A,Xa),B4) ) ) ).

% in_mono
tff(fact_381_subsetD,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,member(A,C2),A3)
       => aa(set(A),$o,member(A,C2),B4) ) ) ).

% subsetD
tff(fact_382_equalityE,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3) ) ) ).

% equalityE
tff(fact_383_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
    <=> ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),A3)
         => aa(set(A),$o,member(A,X3),B4) ) ) ).

% subset_eq
tff(fact_384_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ).

% equalityD1
tff(fact_385_Set_OequalityD2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3) ) ).

% Set.equalityD2
tff(fact_386_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
    <=> ! [T: A] :
          ( aa(set(A),$o,member(A,T),A3)
         => aa(set(A),$o,member(A,T),B4) ) ) ).

% subset_iff
tff(fact_387_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A3) ).

% subset_refl
tff(fact_388_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
         => aa(A,$o,Q,X) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ) ).

% Collect_mono
tff(fact_389_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3) ) ) ).

% subset_trans
tff(fact_390_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3) ) ) ).

% set_eq_subset
tff(fact_391_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))),A3) ).

% Collect_subset
tff(fact_392_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),B4)) ) ).

% less_eq_set_def
tff(fact_393_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q))
    <=> ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(A,$o,Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_394_subset__Collect__conv,axiom,
    ! [A: $tType,S3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),aa(fun(A,$o),set(A),collect(A),P))
    <=> ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),S3)
         => aa(A,$o,P,X3) ) ) ).

% subset_Collect_conv
tff(fact_395_prod__decode__aux_Ocases,axiom,
    ! [Xa: product_prod(nat,nat)] :
      ~ ! [K2: nat,M2: nat] : Xa != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M2) ).

% prod_decode_aux.cases
tff(fact_396_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_397_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_398_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_399_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xa) ).

% vebt_member.simps(2)
tff(fact_400_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_401_filter__preserves__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ao(fun(A,nat),fun(A,$o),M6)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M6),P))) ) ).

% filter_preserves_multiset
tff(fact_402_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Mi) ).

% vebt_mint.simps(3)
tff(fact_403_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Ma) ).

% vebt_maxt.simps(3)
tff(fact_404_vebt__pred_Osimps_I5_J,axiom,
    ! [Va2: 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)),Va2),zero_zero(nat),Vd,Ve),Vf) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_405_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_406_add__mset__in__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),A2: A] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ao(fun(A,nat),fun(A,$o),M6)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aq(fun(A,nat),fun(A,fun(A,$o)),M6),A2))) ) ).

% add_mset_in_multiset
tff(fact_407_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_408_vebt__pred_Osimps_I6_J,axiom,
    ! [Va2: product_prod(nat,nat),Vha: list(vEBT_VEBT),Via: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vha,Via),Vj) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_409_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_410_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_411_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A2: A,B2: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,F),aa(A,option(A),some(A),A2)),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_412_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ar(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_413_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_as(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_414_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_at(set(A),fun(set(A),$o),A3))) ) ).

% finite_Collect_subsets
tff(fact_415_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] : Xa != vEBT_Leaf((A4),(B3))
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_416_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        | aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) )
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_au(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_417_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_av(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% finite_Collect_disjI
tff(fact_418_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,
    ! [Xa: vEBT_VEBT] :
      ( ( Xa != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_419_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X5: set(A)] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
               => ? [Xa2: A] :
                    ( aa(set(A),$o,member(A,Xa2),X5)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa2) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).

% infinite_growing
tff(fact_420_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] :
                ( aa(set(A),$o,member(A,X),S3)
                & ~ ? [Xa2: A] :
                      ( aa(set(A),$o,member(A,Xa2),S3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_421_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv2: option(A)] : Xa != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,A)),V2: A] : Xa != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))
       => ~ ! [F2: fun(A,fun(A,A)),A4: A,B3: A] : Xa != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),F2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A4)),aa(A,option(A),some(A),B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_422_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,$o)),Uv2: option(A)] : Xa != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,$o)),V2: A] : Xa != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))
       => ~ ! [F2: fun(A,fun(A,$o)),X: A,Y3: A] : Xa != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),F2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_423_VEBT__internal_Oreplicatei_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(nat,heap_Time_Heap(A))] :
      ( ! [X: heap_Time_Heap(A)] : Xa != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),zero_zero(nat)),X)
     => ~ ! [N: nat,X: heap_Time_Heap(A)] : Xa != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),aa(nat,nat,suc,N)),X) ) ).

% VEBT_internal.replicatei.cases
tff(fact_424_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),D2)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Deg3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_425_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X: A] :
          ( aa(A,$o,P,X)
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,X)),aa(A,nat,M,Y4)) ) ) ) ).

% ex_has_least_nat
tff(fact_426_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $o,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),X)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList2: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S)),X) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_427_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))
     => ( ! [A4: $o,Uw2: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B3: $o,Va: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT,Vb2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Vb2)
           => ( ! [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2)),Vf2)
             => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_428_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,B3: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))
     => ( ! [Uv2: $o,Uw2: $o,N: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Va3)
         => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)),Ve2)
           => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Vi)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_429_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,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $o,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),X)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S)),X)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S)),X)
         => ( ! [V2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2)),X)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_430_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,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $o,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),X)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),X)
       => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)),X)
         => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_431_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),R: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => ? [Xa2: B] :
                  ( aa(set(B),$o,member(B,Xa2),B4)
                  & aa(B,$o,aa(A,fun(B,$o),R,X),Xa2) ) )
         => ? [X: B] :
              ( aa(set(B),$o,member(B,X),B4)
              & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_aw(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_432_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% not_finite_existsD
tff(fact_433_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Uw2)
     => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),X)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2)),X)
           => ~ ! [V2: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2)),X) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_434_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => ? [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2)
                & ! [Xa2: A] :
                    ( aa(set(A),$o,member(A,Xa2),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_435_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => ? [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
                & ! [Xa2: A] :
                    ( aa(set(A),$o,member(A,Xa2),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_436_infinite__imp__nonempty,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S3)
     => ( S3 != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_437_finite_OemptyI,axiom,
    ! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).

% finite.emptyI
tff(fact_438_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,Y3)),B2) )
       => ? [X: A] :
            ( aa(A,$o,P,X)
            & ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,Y4)),aa(A,nat,F,X)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_439_rev__finite__subset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% rev_finite_subset
tff(fact_440_infinite__super,axiom,
    ! [A: $tType,S3: set(A),T2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
     => ( ~ aa(set(A),$o,finite_finite2(A),S3)
       => ~ aa(set(A),$o,finite_finite2(A),T2) ) ) ).

% infinite_super
tff(fact_441_finite__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,finite_finite2(A),B4)
       => aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% finite_subset
tff(fact_442_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
                & ! [Xa2: A] :
                    ( aa(set(A),$o,member(A,Xa2),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_443_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
                & ! [Xa2: A] :
                    ( aa(set(A),$o,member(A,Xa2),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_444_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,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),zero_zero(nat))
     => ( ! [A4: $o,B3: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B3: $o,N: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))
         => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),X)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),X)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_445_predicate1I,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
         => aa(A,$o,Q,X) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).

% predicate1I
tff(fact_446_vebt__maxt_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  (B3),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_447_vebt__mint_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  (A4),
                  aa(nat,option(nat),some(nat),zero_zero(nat)),
                  $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_448_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S3: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),S3))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R),S3) ) ).

% pred_subset_eq
tff(fact_449_finite__less__ub,axiom,
    ! [F: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,F,N))
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ax(fun(nat,nat),fun(nat,fun(nat,$o)),F),U))) ) ).

% finite_less_ub
tff(fact_450_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ay(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_451_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [Xa: B] : aa(B,A,bot_bot(fun(B,A)),Xa) = bot_bot(A) ) ).

% bot_apply
tff(fact_452_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% dual_order.refl
tff(fact_453_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Xa) ) ).

% order_refl
tff(fact_454_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_455_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg1Leaf
tff(fact_456_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ).

% deg_1_Leaf
tff(fact_457_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_458_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_459_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_460_false__rule,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Q: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),C2,Q) ).

% false_rule
tff(fact_461_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = bot_bot(assn) )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_462_pure__false,axiom,
    pure_assn($false) = bot_bot(assn) ).

% pure_false
tff(fact_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_less__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),one_one(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_one
tff(fact_472_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xa: A] :
          ( ( one_one(A) = Xa )
        <=> ( Xa = one_one(A) ) ) ) ).

% one_reorient
tff(fact_473_bot__empty__eq,axiom,
    ! [A: $tType,X4: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X4)
    <=> aa(set(A),$o,member(A,X4),bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_474_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_475_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_476_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_477_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_478_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_479_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_480_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_481_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_482_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_483_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_az(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_484_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_485_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_486_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_487_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_488_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_489_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Nb)
           => aa(A,$o,aa(A,fun(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_490_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_491_nat__geq__1__eq__neqz,axiom,
    ! [Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Xa)
    <=> ( Xa != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_492_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_493_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_494_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_495_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_496_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% order_class.order_eq_iff
tff(fact_497_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_498_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_499_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
           => ( Xa = Y ) ) ) ) ).

% order_antisym
tff(fact_500_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% order.trans
tff(fact_501_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% order_trans
tff(fact_502_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
         => ( ! [A4: A,B3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
               => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
           => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_503_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_504_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_505_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% dual_order.trans
tff(fact_506_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_507_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_508_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,F,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_subst1
tff(fact_509_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_510_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_eq_refl
tff(fact_511_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_linear
tff(fact_512_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,F,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_513_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F,B2) = C2 )
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_514_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_le_cases
tff(fact_515_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_516_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2) ) ) ) ).

% mult_left_le
tff(fact_517_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_518_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)),Xa) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_519_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa)),Xa) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_520_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xa) ) ).

% lt_ex
tff(fact_521_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] :
        ? [X_13: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X_13) ) ).

% gt_ex
tff(fact_522_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ? [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).

% dense
tff(fact_523_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% less_imp_neq
tff(fact_524_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order.asym
tff(fact_525_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_526_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_527_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X)
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_528_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv3
tff(fact_529_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ( Xa != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_cases
tff(fact_530_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% dual_order.asym
tff(fact_531_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% dual_order.irrefl
tff(fact_532_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
        <=> ? [N6: A] :
              ( aa(A,$o,P,N6)
              & ! [M5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M5),N6)
                 => ~ aa(A,$o,P,M5) ) ) ) ) ).

% exists_least_iff
tff(fact_533_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
           => ( ! [A4: A,B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_534_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans
tff(fact_535_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
            | ( Xa = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_536_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_537_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_538_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_539_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE
tff(fact_540_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_asym
tff(fact_541_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neq_iff
tff(fact_542_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order_less_asym'
tff(fact_543_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_trans
tff(fact_544_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X)),aa(B,A,F,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_545_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F,B2) = C2 )
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_546_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Xa) ) ).

% order_less_irrefl
tff(fact_547_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X)),aa(B,A,F,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_548_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_549_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_not_sym
tff(fact_550_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_551_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          | ( Xa = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_less_linear
tff(fact_552_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_553_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Y != Xa ) ) ) ).

% order_less_imp_not_eq2
tff(fact_554_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_imp_not_less
tff(fact_555_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X: A,Y3: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)),R2)
         => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)),S2) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2) ) ).

% subrelI
tff(fact_556_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X4: A] : aa(A,B,bot_bot(fun(A,B)),X4) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_557_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$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_558_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),Xa: nat,M6: nat] :
      ( aa(nat,$o,P,Xa)
     => ( ! [X: nat] :
            ( aa(nat,$o,P,X)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M6) )
       => ~ ! [M2: nat] :
              ( aa(nat,$o,P,M2)
             => ~ ! [X4: nat] :
                    ( aa(nat,$o,P,X4)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_559_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),Xa)
    <=> $ite(
          Xa = zero_zero(nat),
          (A2),
          $ite(Xa = one_one(nat),(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_560_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F) ) ) ) ).

% less_fun_def
tff(fact_561_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),Xa)
    <=> $ite(
          Xa = zero_zero(nat),
          (A2),
          $ite(Xa = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_562_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Xa: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,Xa)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
       => aa(A,$o,Q,Xa) ) ) ).

% rev_predicate1D
tff(fact_563_predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xa: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
     => ( aa(A,$o,P,Xa)
       => aa(A,$o,Q,Xa) ) ) ).

% predicate1D
tff(fact_564_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_565_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_566_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_567_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_568_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_569_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_570_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_571_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_572_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o))),S3))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S3) ) ).

% pred_subset_eq2
tff(fact_573_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X3: A,Xa3: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),R)
        <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_574_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_575_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_576_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_577_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% leD
tff(fact_578_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% leI
tff(fact_579_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_580_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv1
tff(fact_581_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv2
tff(fact_582_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_583_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_584_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% less_le_not_le
tff(fact_585_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% not_le_imp_less
tff(fact_586_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_587_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_588_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans1
tff(fact_589_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_590_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_591_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Xa)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W2) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_592_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),W2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_593_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_594_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_595_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_596_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_597_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_598_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_599_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_600_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_less
tff(fact_601_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ( Xa != Y ) ) ) ) ).

% order_less_le
tff(fact_602_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_not_le
tff(fact_603_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_not_less
tff(fact_604_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_less_imp_le
tff(fact_605_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( A2 != B2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_606_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_607_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_le_less_trans
tff(fact_608_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_le_trans
tff(fact_609_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X)),aa(B,A,F,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_610_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_611_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,F,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_612_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_613_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_le_less_linear
tff(fact_614_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_615_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A2) ) ).

% bot.extremum
tff(fact_616_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_617_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_618_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_619_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A2) ) ) ).

% bot.not_eq_extremum
tff(fact_620_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_621_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funD
tff(fact_622_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funE
tff(fact_623_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G) ) ) ).

% le_funI
tff(fact_624_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
        <=> ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,G,X3)) ) ) ).

% le_fun_def
tff(fact_625_bounded__nat__set__is__finite,axiom,
    ! [N7: set(nat),Nb: nat] :
      ( ! [X: nat] :
          ( aa(set(nat),$o,member(nat,X),N7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Nb) )
     => aa(set(nat),$o,finite_finite2(nat),N7) ) ).

% bounded_nat_set_is_finite
tff(fact_626_finite__nat__set__iff__bounded,axiom,
    ! [N7: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N7)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),N7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_627_finite__nat__set__iff__bounded__le,axiom,
    ! [N7: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N7)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),N7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_628_vebt__maxti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_maxti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ah(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib),one_one(nat)) ).

% vebt_maxti_hT
tff(fact_629_vebt__minti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_minti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ak(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib),one_one(nat)) ).

% vebt_minti_hT
tff(fact_630_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),Xa)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_631_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xa: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bc(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_632_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_633_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,
    ! [Va2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: 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)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_634_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),Xa) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_635_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),Xa) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_636_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,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xa) = Y )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uv2: $o] : Xa = vEBT_Leaf($true,(Uv2))
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu2: $o] : Xa = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa = 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),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_637_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,nat)] : size_option(A,Xa,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_638_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [Va2: product_prod(nat,nat),Vga: list(vEBT_VEBT),Vha: vEBT_VEBT,Via: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vga,Vha),Via) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_639_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X: A,Y3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X),Y3)
         => aa(B,$o,aa(A,fun(B,$o),Q,X),Y3) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).

% predicate2I
tff(fact_640_pure__true,axiom,
    pure_assn($true) = one_one(assn) ).

% pure_true
tff(fact_641_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = one_one(assn) )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_642_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_643_norm__pre__pure__iff__sng,axiom,
    ! [A: $tType,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,pure_assn((B2)),F,Q)
    <=> ( (B2)
       => hoare_hoare_triple(A,one_one(assn),F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_644_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
     => ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Y)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Y) ) ) ).

% predicate2D
tff(fact_645_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xa: A,Y: B,Q: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Y)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Y) ) ) ).

% rev_predicate2D
tff(fact_646_bot2E,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),Xa),Y) ).

% bot2E
tff(fact_647_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_648_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_649_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_650_norm__pre__pure__rule2,axiom,
    ! [A: $tType,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B2)
       => hoare_hoare_triple(A,one_one(assn),F,Q) )
     => hoare_hoare_triple(A,pure_assn((B2)),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_651_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [X_13: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X_13) ) ).

% linordered_field_no_ub
tff(fact_652_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X4) ) ).

% linordered_field_no_lb
tff(fact_653_minNull__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_N_u_l_l(Ta)),one_one(nat)) ).

% minNull_bound
tff(fact_654_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_655_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_656_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_657_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,Xa: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A2),(B2)),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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),Vc: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_668_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_669_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,Xa: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_670_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_671_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_672_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [Va2: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),zero_zero(nat),Vc,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_673_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_674_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_675_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_676_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_677_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,
    ! [Va2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xa: 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)),Va2),zero_zero(nat),Uy,Uz),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_678_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),Xa) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_679_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),Xa) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_680_builupi_Hcorr,axiom,
    ! [Nb: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))) ).

% builupi'corr
tff(fact_681_builupicorr,axiom,
    ! [Nb: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))) ).

% builupicorr
tff(fact_682_norm__pre__pure__iff__htt,axiom,
    ! [A: $tType,P: assn,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F,Q,Ta)
    <=> ( (B2)
       => time_htt(A,P,F,Q,Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_683_norm__pre__pure__iff__htt_H,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((B2))),P),F,Q,Ta)
    <=> ( (B2)
       => time_htt(A,P,F,Q,Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_684_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_685_htt__htD,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,P,C2,Q,Ta)
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% htt_htD
tff(fact_686_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [Va2: product_prod(nat,nat),Vga: list(vEBT_VEBT),Vha: vEBT_VEBT,Via: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vga,Vha),Via) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_687_less__by__empty,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),B4: set(product_prod(A,A))] :
      ( ( A3 = bot_bot(set(product_prod(A,A))) )
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A3),B4) ) ).

% less_by_empty
tff(fact_688_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [Va2: product_prod(nat,nat),Vha: list(vEBT_VEBT),Via: 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)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vha,Via),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_689_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [Va2: product_prod(nat,nat),Vha: list(vEBT_VEBT),Via: 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)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vha,Via),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_690_htt__vebt__buildupi,axiom,
    ! [Nb: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),vEBT_V441764108873111860ildupi(Nb)) ).

% htt_vebt_buildupi
tff(fact_691_htt__vebt__buildupi_H,axiom,
    ! [Nb: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),vEBT_V441764108873111860ildupi(Nb)) ).

% htt_vebt_buildupi'
tff(fact_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_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_700_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [Va2: 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)),Va2),zero_zero(nat),Vd,Ve),Vf) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_701_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [Va2: 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)),Va2),zero_zero(nat),Vd,Ve),Vf) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_702_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [Va2: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Va2),zero_zero(nat),Vc,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_703_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun(nat,$o),M: nat] :
      ( ! [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2)
         => aa(nat,$o,P,K2) )
     => ( ! [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => ( ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),I4)
                 => aa(nat,$o,P,I4) )
             => aa(nat,$o,P,K2) ) )
       => aa(nat,$o,P,M) ) ) ).

% nat_descend_induct
tff(fact_704_T__vebt__buildupi,axiom,
    ! [Nb: nat,H: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Nb),H)),vEBT_V441764108873111860ildupi(Nb)) ).

% T_vebt_buildupi
tff(fact_705_vebt__buildupi__refines,axiom,
    ! [Nb: nat] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_buildupi(Nb),vEBT_V739175172307565963ildupi(Nb)) ).

% vebt_buildupi_refines
tff(fact_706_TBOUND__vebt__buildupi,axiom,
    ! [Nb: nat] : time_TBOUND(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Nb),vEBT_V441764108873111860ildupi(Nb)) ).

% TBOUND_vebt_buildupi
tff(fact_707_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S3)
    <=> ! [M5: nat] :
        ? [N6: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N6)
          & aa(set(nat),$o,member(nat,N6),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_708_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set(nat)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M2)
         => ? [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N2)
              & aa(set(nat),$o,member(nat,N2),S3) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S3) ) ).

% unbounded_k_infinite
tff(fact_709_infinite__nat__iff__unbounded,axiom,
    ! [S3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S3)
    <=> ! [M5: nat] :
        ? [N6: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N6)
          & aa(set(nat),$o,member(nat,N6),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_710_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] : Xa != vEBT_Leaf((A4),(B3))
     => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa != vEBT_Node(Uu2,Deg2,TreeList2,Summary2) ) ).

% VEBT_internal.height.cases
tff(fact_711_TBOUND__vebt__minti,axiom,
    ! [Ta: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_minti(Ta),one_one(nat)) ).

% TBOUND_vebt_minti
tff(fact_712_TBOUND__vebt__maxti,axiom,
    ! [Ta: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_maxti(Ta),one_one(nat)) ).

% TBOUND_vebt_maxti
tff(fact_713_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ~ ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),S3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X4)),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S3))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_714_refines__case__VEBTi,axiom,
    ! [A: $tType,Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun($o,fun($o,heap_Time_Heap(A))),F12: fun($o,fun($o,heap_Time_Heap(A))),F22: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),F23: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))))] :
      ( ( Tib = Ti )
     => ( ! [A4: $o,B3: $o] : refine_Imp_refines(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F1,(A4)),(B3)),aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F12,(A4)),(B3)))
       => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeArray: array(vEBT_VEBTi),Summary2: vEBT_VEBTi] : refine_Imp_refines(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F22,Info2),Deg2),TreeArray),Summary2),aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F23,Info2),Deg2),TreeArray),Summary2))
         => refine_Imp_refines(A,vEBT_case_VEBTi(heap_Time_Heap(A),F22,F1,Tib),vEBT_case_VEBTi(heap_Time_Heap(A),F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_715_TBOUNDD,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Ta: nat,H: heap_ext(product_unit)] :
      ( time_TBOUND(A,M,Ta)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H)),Ta) ) ).

% TBOUNDD
tff(fact_716_TBOUNDI,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Ta: nat] :
      ( ! [H2: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H2)),Ta)
     => time_TBOUND(A,M,Ta) ) ).

% TBOUNDI
tff(fact_717_TBOUND__def,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Ta: nat] :
      ( time_TBOUND(A,M,Ta)
    <=> ! [H3: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H3)),Ta) ) ).

% TBOUND_def
tff(fact_718_TBOUND__mono,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Ta: nat,T3: nat] :
      ( time_TBOUND(A,C2,Ta)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ta),T3)
       => time_TBOUND(A,C2,T3) ) ) ).

% TBOUND_mono
tff(fact_719_httI__TBOUND,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( time_TBOUND(A,C2,Ta)
       => time_htt(A,P,C2,Q,Ta) ) ) ).

% httI_TBOUND
tff(fact_720_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => aa(set(A),$o,member(A,lattic7623131987881927897min_on(A,B,F,S3)),S3) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_721_finite__transitivity__chain,axiom,
    ! [A: $tType,A3: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X),X)
       => ( ! [X: A,Y3: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R,X),Y3)
             => ( aa(A,$o,aa(A,fun(A,$o),R,Y3),Z3)
               => aa(A,$o,aa(A,fun(A,$o),R,X),Z3) ) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => ? [Y4: A] :
                    ( aa(set(A),$o,member(A,Y4),A3)
                    & aa(A,$o,aa(A,fun(A,$o),R,X),Y4) ) )
           => ( A3 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_722_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),Y: A,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,member(A,Y),S3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S3))),aa(A,B,F,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_723_refines__replicate,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),F3: heap_Time_Heap(A),Nb: nat] :
      ( refine_Imp_refines(A,F,F3)
     => refine_Imp_refines(list(A),vEBT_VEBT_replicatei(A,Nb,F),vEBT_VEBT_replicatei(A,Nb,F3)) ) ).

% refines_replicate
tff(fact_724_hoare__triple__refines,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),C4: heap_Time_Heap(A)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( refine_Imp_refines(A,C4,C2)
       => hoare_hoare_triple(A,P,C4,Q) ) ) ).

% hoare_triple_refines
tff(fact_725_vebt__maxt_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      aa(nat,option(nat),some(nat),one_one(nat)),
                      $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma2) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_726_height__node,axiom,
    ! [Mi: 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary))) ) ).

% height_node
tff(fact_727_vebt__mint_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      (A4),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi2) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_728_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X: A] : ~ aa(set(A),$o,member(A,X),A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_729_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,P,A2)
           => ( ~ aa(A,$o,P,B2)
             => ? [C5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),B2)
                  & ! [X4: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),C5) )
                     => aa(A,$o,P,X4) )
                  & ! [D3: A] :
                      ( ! [X: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D3) )
                         => aa(A,$o,P,X) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C5) ) ) ) ) ) ) ).

% complete_interval
tff(fact_730_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A6: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B6),A6)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A6),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_731_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Ta) ) ) ).

% pinf(6)
tff(fact_732_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X4) ) ) ).

% pinf(8)
tff(fact_733_vebt__memberi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xa: nat,Ta: vEBT_VEBT] : refine_Imp_refines($o,vEBT_vebt_memberi(Tib,Xa),vEBT_V854960066525838166emberi(Ta,Tib,Xa)) ).

% vebt_memberi_refines
tff(fact_734_VEBTi_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),F22: fun($o,fun($o,B)),VEBTi: vEBT_VEBTi] : aa(B,A,H,vEBT_case_VEBTi(B,F1,F22,VEBTi)) = vEBT_case_VEBTi(A,aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_bd(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),H),F1),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_be(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H),F22),VEBTi) ).

% VEBTi.case_distrib
tff(fact_735_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_736_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_737_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% verit_la_disequality
tff(fact_738_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_739_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X4) ) ) ).

% minf(7)
tff(fact_740_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Ta) ) ) ).

% minf(5)
tff(fact_741_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( X4 != Ta ) ) ) ).

% minf(4)
tff(fact_742_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( X4 != Ta ) ) ) ).

% minf(3)
tff(fact_743_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q2,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
               => ( ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    | aa(A,$o,Q2,X4) ) ) ) ) ) ) ).

% minf(2)
tff(fact_744_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q2,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
               => ( ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    & aa(A,$o,Q2,X4) ) ) ) ) ) ) ).

% minf(1)
tff(fact_745_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_746_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X4) ) ) ).

% pinf(7)
tff(fact_747_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Ta) ) ) ).

% pinf(5)
tff(fact_748_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( X4 != Ta ) ) ) ).

% pinf(4)
tff(fact_749_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( X4 != Ta ) ) ) ).

% pinf(3)
tff(fact_750_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q2,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
               => ( ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    | aa(A,$o,Q2,X4) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_751_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q2,X) ) )
           => ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
               => ( ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P2,X4)
                    & aa(A,$o,Q2,X4) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_752_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% verit_comp_simplify1(1)
tff(fact_753_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B3)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A2) ) ) ).

% ex_gt_or_lt
tff(fact_754_prop__restrict,axiom,
    ! [A: $tType,Xa: A,Z5: set(A),X5: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,Xa),Z5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z5),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),X5),P)))
       => aa(A,$o,P,Xa) ) ) ).

% prop_restrict
tff(fact_755_Collect__restrict,axiom,
    ! [A: $tType,X5: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),X5),P))),X5) ).

% Collect_restrict
tff(fact_756_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X4) ) ) ).

% minf(8)
tff(fact_757_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Ta) ) ) ).

% minf(6)
tff(fact_758_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,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one(nat) )
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv2)) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xa = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_759_delete__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% delete_bound_height'
tff(fact_760_time__replicate,axiom,
    ! [A: $tType,Xa: heap_Time_Heap(A),C2: nat,Nb: nat,H: heap_ext(product_unit)] :
      ( ! [H2: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,Xa,H2)),C2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(list(A),vEBT_VEBT_replicatei(A,Nb,Xa),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_761_TBOUND__minNulli,axiom,
    ! [Ta: vEBT_VEBTi] : time_TBOUND($o,vEBT_VEBT_minNulli(Ta),one_one(nat)) ).

% TBOUND_minNulli
tff(fact_762_TBOUND__replicate,axiom,
    ! [A: $tType,Xa: heap_Time_Heap(A),C2: nat,Nb: nat] :
      ( time_TBOUND(A,Xa,C2)
     => time_TBOUND(list(A),vEBT_VEBT_replicatei(A,Nb,Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),C2)),Nb))) ) ).

% TBOUND_replicate
tff(fact_763_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summary))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ).

% height_compose_summary
tff(fact_764_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,$o)),R22: fun(A,fun(A,$o))] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R1),R22)
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),accp(A,R22)),accp(A,R1)) ) ).

% accp_subset
tff(fact_765_accp__subset__induct,axiom,
    ! [A: $tType,D4: fun(A,$o),R: fun(A,fun(A,$o)),Xa: A,P: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),D4),accp(A,R))
     => ( ! [X: A,Z3: A] :
            ( aa(A,$o,D4,X)
           => ( aa(A,$o,aa(A,fun(A,$o),R,Z3),X)
             => aa(A,$o,D4,Z3) ) )
       => ( aa(A,$o,D4,Xa)
         => ( ! [X: A] :
                ( aa(A,$o,D4,X)
               => ( ! [Z4: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),R,Z4),X)
                     => aa(A,$o,P,Z4) )
                 => aa(A,$o,P,X) ) )
           => aa(A,$o,P,Xa) ) ) ) ) ).

% accp_subset_induct
tff(fact_766_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_767_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Ta)
     => ( vEBT_vebt_mint(Ta) = none(nat) ) ) ).

% minNullmin
tff(fact_768_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)
     => ~ ! [N: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ).

% even_odd_cases
tff(fact_769_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => ~ aa(nat,$o,vEBT_vebt_member(Ta),Xa) ) ).

% min_Null_member
tff(fact_770_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none(nat) )
     => vEBT_VEBT_minNull(Ta) ) ).

% minminNull
tff(fact_771_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_772_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_773_minNrulli__ruleT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_minNulli(Tib),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_bf(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Ta),Tib),one_one(nat)) ).

% minNrulli_ruleT
tff(fact_774_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_775_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_776_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_777_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_778_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_779_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_780_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_781_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_782_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_783_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_784_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_785_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_786_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_787_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_788_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_789_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_790_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_791_minNulli__rule,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_minNulli(Tib),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_bf(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Ta),Tib)) ).

% minNulli_rule
tff(fact_792_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_793_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_794_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_795_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_796_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_797_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_798_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_799_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_800_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_801_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_802_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_803_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_804_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_805_add__gr__0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% add_gr_0
tff(fact_806_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_807_TBOUND__minNull,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xa: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => time_TBOUND(vEBT_VEBTi,vEBT_V3964819847710782039nserti(Ta,Tib,Xa),one_one(nat)) ) ).

% TBOUND_minNull
tff(fact_808_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_809_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_810_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_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_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_822_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_823_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_824_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_825_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D)) ) ) ) ).

% add_mono
tff(fact_826_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_left_mono
tff(fact_827_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ~ ! [C5: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) ) ) ).

% less_eqE
tff(fact_828_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_right_mono
tff(fact_829_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ? [C6: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C6) ) ) ).

% le_iff_add
tff(fact_830_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_831_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_832_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(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_833_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,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(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_834_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(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_835_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D)
           => aa(A,$o,aa(A,fun(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_836_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(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_837_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(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_838_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_839_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_840_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_841_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_842_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_843_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_844_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_845_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E2: 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),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),C2) ) ).

% combine_common_factor
tff(fact_846_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_847_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_848_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_849_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_850_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_851_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K) ) ).

% add_lessD1
tff(fact_852_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$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_853_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I) ).

% not_add_less1
tff(fact_854_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_855_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$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_856_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_less_add1
tff(fact_857_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_less_add2
tff(fact_858_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% less_add_eq_less
tff(fact_859_add__leE,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% add_leE
tff(fact_860_le__add1,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)) ).

% le_add1
tff(fact_861_le__add2,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) ).

% le_add2
tff(fact_862_add__leD1,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% add_leD1
tff(fact_863_add__leD2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% add_leD2
tff(fact_864_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_865_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_866_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_867_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_868_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_869_nat__le__iff__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_870_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_871_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_872_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_873_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_874_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_875_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_876_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_877_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing
tff(fact_878_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_879_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing2
tff(fact_880_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_881_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_882_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_883_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_884_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_885_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(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_886_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ ! [C5: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) )
               => ( C5 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_887_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% pos_add_strict
tff(fact_888_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_889_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(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_890_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(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_891_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D)
           => aa(A,$o,aa(A,fun(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_892_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D)
           => aa(A,$o,aa(A,fun(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_893_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(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_894_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(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_895_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_896_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_897_less__imp__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_898_less__iff__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_899_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$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_900_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$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_901_less__natE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => ~ ! [Q3: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_902_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$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_903_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_904_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_905_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_906_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ? [Y4: A] :
                ( aa(A,$o,P,Y4)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,Y4)),aa(A,nat,F,X)) ) )
       => ? [Y3: A] :
            ( aa(A,$o,P,Y3)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,K)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_907_mono__nat__linear__lb,axiom,
    ! [F: fun(nat,nat),M: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F,M2)),aa(nat,nat,F,N)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F,M)),K)),aa(nat,nat,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))) ) ).

% mono_nat_linear_lb
tff(fact_908_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_909_mlex__snd__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,B6: nat,N7: nat] :
      ( ( A2 = A6 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),B6)
       => aa(nat,$o,aa(nat,fun(nat,$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),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N7)),B6)) ) ) ).

% mlex_snd_decrI
tff(fact_910_mlex__fst__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,N7: nat,B6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A6)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),N7)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B6),N7)
         => aa(nat,$o,aa(nat,fun(nat,$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),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N7)),B6)) ) ) ) ).

% mlex_fst_decrI
tff(fact_911_mlex__bound,axiom,
    ! [A2: nat,A3: nat,B2: nat,N7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),N7)
       => aa(nat,$o,aa(nat,fun(nat,$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),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)) ) ) ).

% mlex_bound
tff(fact_912_mlex__leI,axiom,
    ! [A2: nat,A6: nat,B2: nat,B6: nat,N7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),A6)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),B6)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N7)),B6)) ) ) ).

% mlex_leI
tff(fact_913_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_914_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_915_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xa: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bg(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bg(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bh(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_916_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_epsilon
tff(fact_917_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_918_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_919_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(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_920_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_921_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(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_922_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_923_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_924_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(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_925_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$o,aa(A,fun(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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_926_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xa)
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_927_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xa)
     => ( ( Xa != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_928_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xa: A,A2: A,Y: A,U: A,Va2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Va2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),Va2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Va2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_929_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xa)
      <=> (Y) )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xa = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xa = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_930_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_931_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_932_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xa: A,A2: A,Y: A,U: A,Va2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Va2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),Va2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(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),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Va2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_933_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X2: A] : size_option(A,Xa,aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_934_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_935_pred__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d2(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% pred_bound_height'
tff(fact_936_succ_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c2(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% succ'_bound_height
tff(fact_937_member__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r2(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% member_bound_height'
tff(fact_938_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: 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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_939_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_940_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,A)),Xaa: option(A),Xb: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Xa),Xaa),Xb) = Y )
     => ( aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Xa),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),Xaa),Xb)))
       => ( ( ( Xaa = none(A) )
           => ( ( Y = none(A) )
             => ~ aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Xa),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Xb))) ) )
         => ( ! [V2: A] :
                ( ( Xaa = aa(A,option(A),some(A),V2) )
               => ( ( Xb = none(A) )
                 => ( ( Y = none(A) )
                   => ~ aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Xa),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))) ) ) )
           => ~ ! [A4: A] :
                  ( ( Xaa = aa(A,option(A),some(A),A4) )
                 => ! [B3: A] :
                      ( ( Xb = aa(A,option(A),some(A),B3) )
                     => ( ( Y = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xa,A4),B3)) )
                       => ~ aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Xa),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A4)),aa(A,option(A),some(A),B3)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_941_minNull__delete__time__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(Ta,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(Ta,Xa)),one_one(nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_942_height__compose__child,axiom,
    ! [Ta: vEBT_VEBT,TreeList: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Deg: nat,Summary: vEBT_VEBT] :
      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,Ta),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ) ).

% height_compose_child
tff(fact_943_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2) ) ) ).

% discrete
tff(fact_944_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_945_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_946_in__measure,axiom,
    ! [A: $tType,Xa: A,Y: A,F: fun(A,nat)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y)),measure(A,F))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,Xa)),aa(A,nat,F,Y)) ) ).

% in_measure
tff(fact_947_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( ! [A4: $o] :
            ( ? [B3: $o] : Xa = vEBT_Leaf((A4),(B3))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_948_delete__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => vEBT_invar_vebt(vEBT_vebt_delete(Ta,Xa),Nb) ) ).

% delete_pres_valid
tff(fact_949_dele__member__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_delete(Ta,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa(nat,$o,vEBT_vebt_member(Ta),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_950_union__eq__empty,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7) = zero_zero(multiset(A)) )
    <=> ( ( M6 = zero_zero(multiset(A)) )
        & ( N7 = zero_zero(multiset(A)) ) ) ) ).

% union_eq_empty
tff(fact_951_empty__eq__union,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7) )
    <=> ( ( M6 = zero_zero(multiset(A)) )
        & ( N7 = zero_zero(multiset(A)) ) ) ) ).

% empty_eq_union
tff(fact_952_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType,Xa: 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)),Xa),Y) )
    <=> ( ( Xa = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_953_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,Xa: multiset(A),Y: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa),Y) = zero_zero(multiset(A)) )
    <=> ( ( Xa = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_954_size__union,axiom,
    ! [A: $tType,M6: multiset(A),N7: 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)),M6),N7)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),M6)),aa(multiset(A),nat,size_size(multiset(A)),N7)) ).

% size_union
tff(fact_955_vebt__inserti__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xa: nat,Ta: vEBT_VEBT] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_inserti(Tib,Xa),vEBT_V3964819847710782039nserti(Ta,Tib,Xa)) ).

% vebt_inserti_refines
tff(fact_956_empty__neutral_I1_J,axiom,
    ! [A: $tType,Xa: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),zero_zero(multiset(A))),Xa) = Xa ).

% empty_neutral(1)
tff(fact_957_empty__neutral_I2_J,axiom,
    ! [A: $tType,Xa: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa),zero_zero(multiset(A))) = Xa ).

% empty_neutral(2)
tff(fact_958_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_959_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_960_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_961_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_962_vebt__inserti_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_V3964819847710782039nserti(Ta,Tib,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Ta,Xa))) ).

% vebt_inserti'_rf_abstr
tff(fact_963_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_964_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_965_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bi(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_le
tff(fact_966_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_967_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q4: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q4),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_968_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bj(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_eq
tff(fact_969_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_970_add__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),Xa)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% add_shift
tff(fact_971_union__assoc,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),K4: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7)),K4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),K4)) ).

% union_assoc
tff(fact_972_union__lcomm,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),K4: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),K4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),K4)) ).

% union_lcomm
tff(fact_973_union__commute,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),M6) ).

% union_commute
tff(fact_974_union__left__cancel,axiom,
    ! [A: $tType,K4: multiset(A),M6: multiset(A),N7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K4),M6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K4),N7) )
    <=> ( M6 = N7 ) ) ).

% union_left_cancel
tff(fact_975_union__right__cancel,axiom,
    ! [A: $tType,M6: multiset(A),K4: multiset(A),N7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),K4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),K4) )
    <=> ( M6 = N7 ) ) ).

% union_right_cancel
tff(fact_976_multi__union__self__other__eq,axiom,
    ! [A: $tType,A3: multiset(A),X5: multiset(A),Y6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),X5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),Y6) )
     => ( X5 = Y6 ) ) ).

% multi_union_self_other_eq
tff(fact_977_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D4: multiset(A),C3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B4),D4)
         => aa(multiset(A),$o,aa(multiset(A),fun(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),C3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D4),C3)) ) ) ).

% union_le_mono1
tff(fact_978_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D4: multiset(A),C3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B4),D4)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),D4)) ) ) ).

% union_le_mono2
tff(fact_979_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: multiset(A),C3: multiset(A),B4: multiset(A),D4: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),A3),C3)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B4),D4)
           => aa(multiset(A),$o,aa(multiset(A),fun(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)),C3),D4)) ) ) ) ).

% union_less_mono
tff(fact_980_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_981_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_982_finite__maxlen,axiom,
    ! [A: $tType,M6: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M6)
     => ? [N: nat] :
        ! [X4: list(A)] :
          ( aa(set(list(A)),$o,member(list(A),X4),M6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X4)),N) ) ) ).

% finite_maxlen
tff(fact_983_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$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_984_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B4)
    <=> ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Xs))
         => aa(set(A),$o,member(A,X3),B4) ) ) ).

% subset_code(1)
tff(fact_985_length__pos__if__in__set,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_986_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_987_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_988_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_989_vebt__insert_Osimps_I4_J,axiom,
    ! [Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_990_vebt__insert_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xa) = $ite(
        Xa = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(Xa = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_991_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2),Xa) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ).

% vebt_insert.simps(3)
tff(fact_992_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts2,S2),Xa) = vEBT_Node(Info,zero_zero(nat),Ts2,S2) ).

% vebt_insert.simps(2)
tff(fact_993_mi__eq__ma__no__ch,axiom,
    ! [Mi: 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
             => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_1) )
          & ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_994_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xa)
      <=> (Y) )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xa = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_995_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_996_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_997_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32))))
     => ( ! [F2: fun(nat,fun(A,A)),A4: nat,B3: nat,Acc: A] :
            ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A4)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B3),aa(A,A,aa(nat,fun(A,A),F2,A4),Acc)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F2),A4),B3),Acc) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_998_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13) ) ).

% not_min_Null_member
tff(fact_999_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xa) )
     => aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa) ) ).

% maxbmo
tff(fact_1000_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_delete(Ta,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_1001_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
      <=> aa(nat,$o,vEBT_vebt_member(Ta),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_1002_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
       => aa(nat,$o,vEBT_vebt_member(Ta),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_1003_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
    <=> ( vEBT_V5719532721284313246member(Ta,Xa)
        | vEBT_VEBT_membermima(Ta,Xa) ) ) ).

% both_member_options_def
tff(fact_1004_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M6),M6) ) ).

% mset_le_not_refl
tff(fact_1005_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M6),N7)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N7),M6) ) ) ).

% mset_le_not_sym
tff(fact_1006_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M6),M6) ) ).

% mset_le_irrefl
tff(fact_1007_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K4: multiset(A),M6: multiset(A),N7: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K4),M6)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M6),N7)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K4),N7) ) ) ) ).

% mset_le_trans
tff(fact_1008_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M6),N7)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N7),M6) ) ) ).

% mset_le_asym
tff(fact_1009_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less_eq(multiset(A)),M6),N7)
        <=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M6),N7)
            | ( M6 = N7 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_1010_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_1011_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_1012_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_1013_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_1014_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_1015_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_1016_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,Xa: 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),Xa),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),Xa),Y)) )
        <=> ( ( W = Xa )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_1017_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xa)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ! [Uv2: $o] :
              ( ( Xa = vEBT_Leaf($true,(Uv2)) )
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xa = vEBT_Leaf((Uu2),$true) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_1018_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xa)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_1019_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_1020_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb,Xc) = Y )
     => ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),Xa),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) )
           => ~ aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),Xa),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_1021_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2))))
     => ( set_fo6178422350223883121st_nat(A,F,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A2),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_1022_foldr__same__int,axiom,
    ! [Xs: list(nat),Y: nat] :
      ( ! [X: nat,Y3: nat] :
          ( aa(set(nat),$o,member(nat,X),aa(list(nat),set(nat),set2(nat),Xs))
         => ( aa(set(nat),$o,member(nat,Y3),aa(list(nat),set(nat),set2(nat),Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: nat] :
            ( aa(set(nat),$o,member(nat,X),aa(list(nat),set(nat),set2(nat),Xs))
           => ( X = Y ) )
       => ( foldr(nat,nat,plus_plus(nat),Xs,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_1023_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,size_size(vEBT_VEBT),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_1024_insert_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% insert'_bound_height
tff(fact_1025_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Nb: nat] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
     => ( aa(nat,$o,aa(nat,fun(nat,$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_1026_set__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_bk(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_1027_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B3),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A4),zero_zero(nat))
       => ( ! [A4: nat,B3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_1028_foldr__one,axiom,
    ! [D: nat,Ys: list(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),D),foldr(nat,nat,plus_plus(nat),Ys,D)) ).

% foldr_one
tff(fact_1029_foldr__mono,axiom,
    ! [Xs: list(nat),Ys: list(nat),C2: nat,D: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xs))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,nth(nat,Xs),I2)),aa(nat,nat,nth(nat,Ys),I2)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),D)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),foldr(nat,nat,plus_plus(nat),Xs,C2)),aa(list(nat),nat,size_size(list(nat)),Ys))),foldr(nat,nat,plus_plus(nat),Ys,D)) ) ) ) ).

% foldr_mono
tff(fact_1030_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : foldr(A,nat,aTP_Lamp_bl(A,fun(nat,nat)),L,zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_1031_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(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) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1032_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
         => ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,I3),X_12) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I3),aa(nat,A,nth(A,Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_1033_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1034_obtain__list__from__elements,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(nat,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P,Li),I2) )
     => ~ ! [L2: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
                 => aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_1035_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),Nb: nat,Xs: list(A)] :
      ( aa(set(list(A)),$o,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_1036_nth__mem,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => aa(set(A),$o,member(A,aa(nat,A,nth(A,Xs),Nb)),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_1037_list__ball__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1038_in__set__conv__nth,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_1039_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xa: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_1040_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$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)) ) ) ).

% all_set_conv_all_nth
tff(fact_1041_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),L))
         => aa(A,$o,P,X3) )
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L))
         => aa(A,$o,P,aa(nat,A,nth(A,L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_1042_size__list__estimation,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),size_list(A,F,Xs)) ) ) ).

% size_list_estimation
tff(fact_1043_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,nat),G: fun(A,nat)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,G,X)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(A,F,Xs)),size_list(A,G,Xs)) ) ).

% size_list_pointwise
tff(fact_1044_size__list__estimation_H,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),size_list(A,F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_1045_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,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A2),(B2)),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_1046_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A2: nat] : foldr(A,nat,aTP_Lamp_bl(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_1047_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)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts2,S2),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_1048_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_1049_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A2),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_1050_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)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_1051_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,
    ! [Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_1052_insersimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_1053_insertsimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_1054_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts2))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_maxti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_bm(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts2),Tsi)) ) ).

% vebt_maxtilist
tff(fact_1055_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts2))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_minti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_bn(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts2),Tsi)) ) ).

% vebt_mintilist
tff(fact_1056_foldr__zero,axiom,
    ! [Xs: list(nat),D: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,nth(nat,Xs),I2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),aa(nat,nat,minus_minus(nat,foldr(nat,nat,plus_plus(nat),Xs,D)),D)) ) ).

% foldr_zero
tff(fact_1057_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,vEBT_size_VEBT,X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_1058_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X)),Bound) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),foldr(nat,nat,plus_plus(nat),map(A,nat,F,Xs),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_1059_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ! [X: list(A)] :
          ( aa(set(list(A)),$o,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_1060_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_bo(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_1061_sorted__in__between,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: nat,J: nat,L: list(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
             => ( sorted_wrt(A,ord_less_eq(A),L)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I)),Xa)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,nth(A,L),J))
                   => ~ ! [K2: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K2)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),J)
                           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),K2)),Xa)
                             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat)))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_1062_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Nb,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_1063_map__ident,axiom,
    ! [A: $tType,X4: list(A)] : map(A,A,aTP_Lamp_bp(A,A),X4) = X4 ).

% map_ident
tff(fact_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1078_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_1079_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_1080_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Nb)
     => ( aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,minus_minus(nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1081_length__map,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),map(B,A,F,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_1082_length__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,Nb,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_1083_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1084_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1085_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1086_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1087_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_1088_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_1089_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1090_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_1091_zero__less__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,minus_minus(nat,Nb),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ).

% zero_less_diff
tff(fact_1092_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_1093_diff__is__0__eq_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,nat,minus_minus(nat,M),Nb) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1094_diff__is__0__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Nb) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% diff_is_0_eq
tff(fact_1095_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1096_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1097_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1098_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_1099_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1100_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1101_Suc__diff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Nb),M)) = aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_1102_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,map(A,B,F,Xs)),Nb) = aa(A,B,F,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_1103_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_1104_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
        <=> sorted_wrt(B,aTP_Lamp_bq(fun(B,A),fun(B,fun(B,$o)),F),Xs) ) ) ).

% sorted_map
tff(fact_1105_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_br(A,fun(A,$o)),Xs) ).

% sorted_wrt_true
tff(fact_1106_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),F: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R,map(B,A,F,Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_bs(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R),F),Xs) ) ).

% sorted_wrt_map
tff(fact_1107_list_Omap__ident,axiom,
    ! [A: $tType,Ta: list(A)] : map(A,A,aTP_Lamp_bp(A,A),Ta) = Ta ).

% list.map_ident
tff(fact_1108_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_1109_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_1110_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_1111_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( map(B,A,F,Xs) = map(C,A,G,Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_1112_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_1113_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,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1114_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2)) ) ) ).

% diff_right_mono
tff(fact_1115_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2)) ) ) ).

% diff_left_mono
tff(fact_1116_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D)) ) ) ) ).

% diff_mono
tff(fact_1117_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(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_1118_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(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_1119_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,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D) ) ) ) ).

% diff_eq_diff_less
tff(fact_1120_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D),C2)
           => aa(A,$o,aa(A,fun(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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
           => ! [X4: A,K5: A] :
                ( ( aa(A,$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),D4)))
                  & aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1131_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
           => ! [X4: A,K5: A] :
                ( ( aa(A,$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),D4)))
                  | aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1139_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_1140_diff__less__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),L)
       => aa(nat,$o,aa(nat,fun(nat,$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_1141_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_1142_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_1143_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_1144_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_1145_diff__le__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,L),Nb)),aa(nat,nat,minus_minus(nat,L),M)) ) ).

% diff_le_mono2
tff(fact_1146_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,C2),A2)),aa(nat,nat,minus_minus(nat,C2),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_1147_diff__le__self,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Nb)),M) ).

% diff_le_self
tff(fact_1148_diff__le__mono,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_1149_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Nb),K)) = aa(nat,nat,minus_minus(nat,M),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1150_le__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% le_diff_iff
tff(fact_1151_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,Nb),K) )
        <=> ( M = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_1152_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_1153_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_1154_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% strict_sorted_imp_sorted
tff(fact_1155_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys)
           => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_1156_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_1157_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A2),B2)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_1158_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,minus_minus(A,Nb),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1159_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Nb),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1160_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_1161_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_1162_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),A2) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1163_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_1164_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,minus_minus(A,B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1165_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B2),A2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1166_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1167_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1168_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1169_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,minus_minus(A,C2),aa(A,A,minus_minus(A,B2),A2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1170_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,minus_minus(A,B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1171_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( ( aa(A,A,minus_minus(A,B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1172_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(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_1173_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_1174_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_1175_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xa: A,Y: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),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),Xa),Y)),aa(A,A,minus_minus(A,Xa),Y)) ) ).

% square_diff_square_factored
tff(fact_1176_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: 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),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),D) ) ) ) ).

% eq_add_iff2
tff(fact_1177_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: 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),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),C2) = D ) ) ) ).

% eq_add_iff1
tff(fact_1178_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xa: A,Y: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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),Xa),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,Xa),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_1179_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_1180_Suc__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_1181_diff__less__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Nb)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_1182_diff__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Nb)),M) ) ) ).

% diff_less
tff(fact_1183_Suc__diff__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Nb) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),Nb)) ) ) ).

% Suc_diff_le
tff(fact_1184_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_1185_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_1186_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_1187_add__diff__inverse__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$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_1188_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),A2)
       => aa(nat,$o,aa(nat,fun(nat,$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_1189_less__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ) ).

% less_diff_iff
tff(fact_1190_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_1191_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1192_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1193_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1194_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,minus_minus(nat,J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1195_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_1196_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_1197_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_1198_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E2)),C2)),D) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1199_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E2)),D)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1200_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D))
        <=> aa(A,$o,aa(A,fun(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)),E2)),C2)),D) ) ) ).

% less_add_iff1
tff(fact_1201_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D))
        <=> aa(A,$o,aa(A,fun(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)),E2)),D)) ) ) ).

% less_add_iff2
tff(fact_1202_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),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),Xa),one_one(A))),aa(A,A,minus_minus(A,Xa),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1203_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_1204_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_1205_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,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D5: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
           => aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split
tff(fact_1206_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,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D5: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
              & ~ aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_1207_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_1208_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_1209_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_1210_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_1211_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_1212_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1213_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_1214_diff__preserves__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),N7: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ao(fun(A,nat),fun(A,$o),M6)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,nat),fun(A,$o),aTP_Lamp_bt(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_1215_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_1216_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_1217_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$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_1218_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_1219_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_1220_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_1221_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$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_1222_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_1223_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$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_1224_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$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_1225_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_1226_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_1227_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_1228_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_1229_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_bu(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_1230_Suc__if__eq,axiom,
    ! [A: $tType,F: fun(nat,A),H: fun(nat,A),G: A,Nb: nat] :
      ( ! [N: nat] : aa(nat,A,F,aa(nat,nat,suc,N)) = aa(nat,A,H,N)
     => ( ( aa(nat,A,F,zero_zero(nat)) = G )
       => ( aa(nat,A,F,Nb) = $ite(Nb = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_1231_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_1232_repli__emp,axiom,
    ! [A: $tType,B: $tType,Xa: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Nb: nat] :
      ( hoare_hoare_triple(A,one_one(assn),Xa,aa(B,fun(A,assn),A3,Y))
     => hoare_hoare_triple(list(A),one_one(assn),vEBT_VEBT_replicatei(A,Nb,Xa),vEBT_List_list_assn(B,A,A3,replicate(B,Nb,Y))) ) ).

% repli_emp
tff(fact_1233_repli__cons__repl,axiom,
    ! [B: $tType,A: $tType,Q: assn,Xa: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Nb: nat] :
      ( hoare_hoare_triple(A,Q,Xa,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_bv(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,Xa),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_bw(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Q),A3),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_1234_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = 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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_1235_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,minus_minus(A,Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% diff_shunt_var
tff(fact_1236_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(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_1237_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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList),zero_zero(nat))) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_1238_intind,axiom,
    ! [A: $tType,I: nat,Nb: nat,P: fun(A,$o),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(A,$o,P,Xa)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Nb,Xa)),I)) ) ) ).

% intind
tff(fact_1239_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1240_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1241_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_1242_diff__diff__add__mset,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),P: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),N7)),P) = aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),P)) ).

% diff_diff_add_mset
tff(fact_1243_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A3),B4) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1244_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,Xa: A,Nb: nat,Y: A] :
      ( ( replicate(A,M,Xa) = replicate(A,Nb,Y) )
    <=> ( ( M = Nb )
        & ( ( M != zero_zero(nat) )
         => ( Xa = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1245_length__replicate,axiom,
    ! [A: $tType,Nb: nat,Xa: A] : aa(list(A),nat,size_size(list(A)),replicate(A,Nb,Xa)) = Nb ).

% length_replicate
tff(fact_1246_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P,X3) )
    <=> ( aa(A,$o,P,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_1247_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P,X3) )
    <=> ( aa(A,$o,P,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_1248_in__set__replicate,axiom,
    ! [A: $tType,Xa: A,Nb: nat,Y: A] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),replicate(A,Nb,Y)))
    <=> ( ( Xa = Y )
        & ( Nb != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_1249_nth__replicate,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xa)),I) = Xa ) ) ).

% nth_replicate
tff(fact_1250_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_1251_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = aa(nat,nat,minus_minus(nat,To),From) ) ) ) ).

% slice_len
tff(fact_1252_le__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & order(A) )
     => ! [Y: A,Xa: A,A2: word(B),C2: word(B),B2: word(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A2),C2)),B2)
           => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),A2),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A2),C2))
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),C2),aa(word(B),word(B),minus_minus(word(B),B2),A2)) ) ) ) ) ).

% le_minus
tff(fact_1253_double__diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
       => ( aa(set(A),set(A),minus_minus(set(A),B4),aa(set(A),set(A),minus_minus(set(A),C3),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1254_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B4)),A3) ).

% Diff_subset
tff(fact_1255_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),D4: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),B4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B4)),aa(set(A),set(A),minus_minus(set(A),C3),D4)) ) ) ).

% Diff_mono
tff(fact_1256_diff__empty,axiom,
    ! [A: $tType,M6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),zero_zero(multiset(A))) = M6 )
      & ( aa(multiset(A),multiset(A),minus_minus(multiset(A),zero_zero(multiset(A))),M6) = zero_zero(multiset(A)) ) ) ).

% diff_empty
tff(fact_1257_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_1258_Multiset_Odiff__add,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),Q)) = aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),N7)),Q) ).

% Multiset.diff_add
tff(fact_1259_diff__union__cancelL,axiom,
    ! [A: $tType,N7: multiset(A),M6: 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)),N7),M6)),N7) = M6 ).

% diff_union_cancelL
tff(fact_1260_diff__union__cancelR,axiom,
    ! [A: $tType,M6: multiset(A),N7: 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)),M6),N7)),N7) = M6 ).

% diff_union_cancelR
tff(fact_1261_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => ( X = Xa ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xa) = Xs ) ) ).

% replicate_length_same
tff(fact_1262_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xa: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y3: A] :
            ( aa(set(A),$o,member(A,Y3),aa(list(A),set(A),set2(A),Xs))
           => ( Y3 = Xa ) )
       => ( Xs = replicate(A,Nb,Xa) ) ) ) ).

% replicate_eqI
tff(fact_1263_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Nb: nat,Xa: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,Nb,Xa)) ) ).

% sorted_replicate
tff(fact_1264_subset__minus__empty,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),set(A),minus_minus(set(A),A3),B4) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_1265_union__diff__assoc,axiom,
    ! [A: $tType,C3: multiset(A),B4: multiset(A),A3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C3),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)),C3) = 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),C3)) ) ) ).

% union_diff_assoc
tff(fact_1266_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : map(B,A,aTP_Lamp_bx(A,fun(B,A),K),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_1267_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S3: set(A),S4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_by(set(A),fun(set(A),fun(set(A),$o)),S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_1268_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M6: multiset(A),M9: multiset(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,aa(multiset(A),nat,size_size(multiset(A)),M6)),aa(multiset(A),nat,size_size(multiset(A)),M9))),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),M9))) ).

% diff_size_le_size_Diff
tff(fact_1269_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_1270_ranI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),M,A2) = aa(A,option(A),some(A),B2) )
     => aa(set(A),$o,member(A,B2),ran(B,A,M)) ) ).

% ranI
tff(fact_1271_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = 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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList2),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_1272_delete__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xa)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(Ta)),aa(set(nat),set(nat),insert(nat,Xa),bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_1273_delete__correct_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xa)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_VEBT_set_vebt(Ta)),aa(set(nat),set(nat),insert(nat,Xa),bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_1274_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_1275_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),map(A,list(product_prod(A,B)),aTP_Lamp_bz(list(B),fun(A,list(product_prod(A,B))),Ys),Xs))) ).

% product_code
tff(fact_1276_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,minus_minus(nat,Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,minus_minus(nat,Xaa),aa(nat,nat,suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_1277_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      nat_prod_decode_aux(K,M) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,minus_minus(nat,K),M)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_1278_diff__diff__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,M),Nb)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb) ) ) ).

% diff_diff_less
tff(fact_1279_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),zero_zero(word(A)))
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_1280_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),zero_zero(word(A)))
        <=> ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_1281_singletonI,axiom,
    ! [A: $tType,A2: A] : aa(set(A),$o,member(A,A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ).

% singletonI
tff(fact_1282_insert__subset,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),A3)),B4)
    <=> ( aa(set(A),$o,member(A,Xa),B4)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ) ).

% insert_subset
tff(fact_1283_singleton__conv,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ca(A,fun(A,$o),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_1284_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_1285_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),insert(A,A2),A3) = aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1286_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) = aa(set(A),set(A),insert(A,A2),A3) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_1287_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = aa(set(A),set(A),insert(A,A2),A3) ).

% insert_Diff_single
tff(fact_1288_map__update__eta__repair_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,Va2: 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_cb(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M),K),Va2)) = aa(set(A),set(A),insert(A,Va2),ran(B,A,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_1289_set__replicate,axiom,
    ! [A: $tType,Nb: nat,Xa: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Nb,Xa)) = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_1290_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)) ) ) ) ).

% word_diff_ls(2)
tff(fact_1291_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_1292_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(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),Xa)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb)) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_1293_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% Word.word_l_diffs(1)
tff(fact_1294_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),Xb: word(A),W: word(A),Xa: word(A),Xc: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xc),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb))
           => aa(word(A),$o,aa(word(A),fun(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)),Va2),Xb)),Xc)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xc)) ) ) ) ).

% word_plus_mcs(2)
tff(fact_1295_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),Xb: word(A),Xa: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb)),Xa)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_plus_mcs(1)
tff(fact_1296_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)),Y) ) ) ) ).

% word_less_add_right
tff(fact_1297_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)),Z) ) ) ) ).

% word_less_sub_right
tff(fact_1298_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C2)) ) ) ) ).

% plus_minus_no_overflow
tff(fact_1299_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_1300_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),K)) ) ) ) ) ).

% word_less_nowrapI
tff(fact_1301_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Xa),Z))
        <=> ( ( Z = zero_zero(word(A)) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Z) ) ) ) ).

% sub_wrap
tff(fact_1302_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa)) ) ) ) ).

% word_less_minus_mono_left
tff(fact_1303_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_1304_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A)))) ) ) ) ).

% word_le_sub1
tff(fact_1305_word__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)),Xa) ) ) ).

% word_sub_le
tff(fact_1306_word__sub__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)),Xa)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa) ) ) ).

% word_sub_le_iff
tff(fact_1307_word__le__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),D: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),C2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),A2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),C2),D)),C2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),aa(word(A),word(A),minus_minus(word(A),C2),D)) ) ) ) ) ) ).

% word_le_minus_mono
tff(fact_1308_word__le__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z) ) ) ) ).

% word_le_minus_cancel
tff(fact_1309_word__le__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa)) ) ) ) ).

% word_le_minus_mono_left
tff(fact_1310_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_1311_plus__minus__no__overflow__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C2)) ) ) ) ).

% plus_minus_no_overflow_ab
tff(fact_1312_le__minus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2)) ) ) ) ).

% le_minus'
tff(fact_1313_le__plus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2) ) ) ) ).

% le_plus'
tff(fact_1314_le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2) ) ) ) ).

% le_plus
tff(fact_1315_word__plus__mcs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),Xb: word(A),Xa: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb)),Xa)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_plus_mcs(3)
tff(fact_1316_word__plus__mcs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),Xb: word(A),W: word(A),Xa: word(A),Xc: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xc),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xb)),Xc)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xc)) ) ) ) ).

% word_plus_mcs(4)
tff(fact_1317_Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% Word.word_l_diffs(3)
tff(fact_1318_Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb)) ) ) ) ).

% Word.word_l_diffs(4)
tff(fact_1319_word__diff__ls_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_diff_ls(3)
tff(fact_1320_word__diff__ls_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)) ) ) ) ).

% word_diff_ls(4)
tff(fact_1321_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_cc(set(A),fun(set(A),fun(A,$o)),A3),B4)) ).

% set_diff_eq
tff(fact_1322_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),minus_minus(fun(A,$o),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),B4))) ).

% minus_set_def
tff(fact_1323_Multiset_Odiff__right__commute,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),N7)),Q) = aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),Q)),N7) ).

% Multiset.diff_right_commute
tff(fact_1324_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),one_one(word(A))),Xa)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A)))) ) ) ) ).

% word_less_sub1
tff(fact_1325_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Xa),Z))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Z) ) ) ).

% sub_wrap_lt
tff(fact_1326_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Xa),Y))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ).

% word_sub_less_iff
tff(fact_1327_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),C2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),D),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),A2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),C2),D)),C2)
               => aa(word(A),$o,aa(word(A),fun(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_1328_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),I)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A))))
              & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ) ) ).

% inc_i
tff(fact_1329_inc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),M)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ).

% inc_le
tff(fact_1330_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y7: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y7),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_1331_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
        <=> ( ( Xa = Y )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_le_less_eq
tff(fact_1332_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),Y) ) ).

% word_zero_le
tff(fact_1333_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),zero_zero(word(A)))
         => ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_1334_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),A2) ) ).

% word_coorder.extremum
tff(fact_1335_word__plus__mono__right2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)) ) ) ) ).

% word_plus_mono_right2
tff(fact_1336_word__plus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)) ) ) ) ).

% word_plus_mono_right
tff(fact_1337_word__le__plus__either,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z)) ) ) ) ).

% word_le_plus_either
tff(fact_1338_le__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) ) ) ) ).

% le_no_overflow
tff(fact_1339_olen__add__eqv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa)) ) ) ).

% olen_add_eqv
tff(fact_1340_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_1341_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_1342_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] : aa(set(A),set(A),insert(A,A2),aa(fun(A,$o),set(A),collect(A),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cd(A,fun(fun(A,$o),fun(A,$o)),A2),P)) ).

% insert_Collect
tff(fact_1343_insert__compr,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : aa(set(A),set(A),insert(A,A2),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ce(A,fun(set(A),fun(A,$o)),A2),B4)) ).

% insert_compr
tff(fact_1344_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y7: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7)),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),Xa)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y7),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_1345_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,aa(word(A),fun(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_1346_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_1347_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_1348_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C2: A,D: A] :
      ( ( aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,C2),aa(set(A),set(A),insert(A,D),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1349_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),A3) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_1350_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) = aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_1351_set__minus__singleton__eq,axiom,
    ! [A: $tType,Xa: A,X5: set(A)] :
      ( ~ aa(set(A),$o,member(A,Xa),X5)
     => ( aa(set(A),set(A),minus_minus(set(A),X5),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_1352_Diff__insert__absorb,axiom,
    ! [A: $tType,Xa: A,A3: set(A)] :
      ( ~ aa(set(A),$o,member(A,Xa),A3)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,Xa),A3)),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_1353_insert__minus__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,A3: set(A)] :
      ( ( Xa != Y )
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,Xa),A3)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,Xa),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_1354_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))),B4) ).

% Diff_insert2
tff(fact_1355_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_1356_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B4)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_1357_insert__subsetI,axiom,
    ! [A: $tType,Xa: A,A3: set(A),X5: set(A)] :
      ( aa(set(A),$o,member(A,Xa),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),X5)),A3) ) ) ).

% insert_subsetI
tff(fact_1358_insert__mono,axiom,
    ! [A: $tType,C3: set(A),D4: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),D4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A2),C3)),aa(set(A),set(A),insert(A,A2),D4)) ) ).

% insert_mono
tff(fact_1359_subset__insert,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B4: set(A)] :
      ( ~ aa(set(A),$o,member(A,Xa),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xa),B4))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ) ).

% subset_insert
tff(fact_1360_subset__insertI,axiom,
    ! [A: $tType,B4: set(A),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(set(A),set(A),insert(A,A2),B4)) ).

% subset_insertI
tff(fact_1361_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),B4)) ) ).

% subset_insertI2
tff(fact_1362_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),Xa: A,C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B4),aa(set(A),set(A),insert(A,Xa),C3)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B4),C3))
        & ~ aa(set(A),$o,member(A,Xa),A3) ) ) ).

% subset_Diff_insert
tff(fact_1363_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cf(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_1364_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_1365_finite_Ocases,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( ( A2 != bot_bot(set(A)) )
       => ~ ! [A7: set(A)] :
              ( ? [A4: A] : A2 = aa(set(A),set(A),insert(A,A4),A7)
             => ~ aa(set(A),$o,finite_finite2(A),A7) ) ) ) ).

% finite.cases
tff(fact_1366_finite_Osimps,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
    <=> ( ( A2 = bot_bot(set(A)) )
        | ? [A8: set(A),A5: A] :
            ( ( A2 = aa(set(A),set(A),insert(A,A5),A8) )
            & aa(set(A),$o,finite_finite2(A),A8) ) ) ) ).

% finite.simps
tff(fact_1367_finite__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ aa(set(A),$o,member(A,X),F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X),F5)) ) ) )
         => aa(set(A),$o,P,F4) ) ) ) ).

% finite_induct
tff(fact_1368_finite__ne__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ( F4 != bot_bot(set(A)) )
       => ( ! [X: A] : aa(set(A),$o,P,aa(set(A),set(A),insert(A,X),bot_bot(set(A))))
         => ( ! [X: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ aa(set(A),$o,member(A,X),F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X),F5)) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_1369_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),A3: set(A)] :
      ( ! [A7: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A7)
         => aa(set(A),$o,P,A7) )
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ aa(set(A),$o,member(A,X),F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X),F5)) ) ) )
         => aa(set(A),$o,P,A3) ) ) ) ).

% infinite_finite_induct
tff(fact_1370_infinite__remove,axiom,
    ! [A: $tType,S3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),S3)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_1371_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A3: set(A)] :
      ( aa(set(A),$o,X5,A3)
     => ( ! [A7: set(A)] :
            ( aa(set(A),$o,X5,A7)
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),A7)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% infinite_coinduct
tff(fact_1372_finite__empty__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,P,A3)
       => ( ! [A4: A,A7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A7)
             => ( aa(set(A),$o,member(A,A4),A7)
               => ( aa(set(A),$o,P,A7)
                 => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),insert(A,A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_1373_subset__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))
    <=> ( ( X5 = bot_bot(set(A)) )
        | ( X5 = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_1374_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_1375_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),Xa: A,B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),B4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xa),B4)) ) ).

% Diff_single_insert
tff(fact_1376_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xa: A,B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xa),B4))
    <=> $ite(aa(set(A),$o,member(A,Xa),A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),B4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)) ) ).

% subset_insert_iff
tff(fact_1377_remove__subset,axiom,
    ! [A: $tType,Xa: A,S3: set(A)] :
      ( aa(set(A),$o,member(A,Xa),S3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),S3) ) ).

% remove_subset
tff(fact_1378_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_bz(list(B),fun(A,list(product_prod(A,B))),Ys),Xs)) ).

% product_concat_map
tff(fact_1379_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),P: fun(set(A),$o),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X: A,S5: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S5)
                 => ( ! [Y4: A] :
                        ( aa(set(A),$o,member(A,Y4),S5)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y4)),aa(A,B,F,X)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X),S5)) ) ) )
             => aa(set(A),$o,P,S3) ) ) ) ) ).

% finite_ranking_induct
tff(fact_1380_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,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A7)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),B3) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1381_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,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A7)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),X4) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1382_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( aa(set(A),$o,member(A,A4),A3)
                 => ( ~ aa(set(A),$o,member(A,A4),F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,A4),F5)) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_1383_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( aa(set(A),$o,member(A,A4),A3)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A3)
                   => ( ~ aa(set(A),$o,member(A,A4),F5)
                     => ( aa(set(A),$o,P,F5)
                       => aa(set(A),$o,P,aa(set(A),set(A),insert(A,A4),F5)) ) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1384_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B4: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite2(A),B4)
         => aa(set(A),$o,P,B4) )
       => ( ! [A7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B4)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B4) ) ) ) ).

% remove_induct
tff(fact_1385_finite__remove__induct,axiom,
    ! [A: $tType,B4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B4)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1386_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)))
       => ( ! [T4: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T4),S3)
             => ( aa(set(A),$o,P,T4)
               => ? [X4: A] :
                    ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),minus_minus(set(A),S3),T4))
                    & aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),T4)) ) ) )
         => aa(set(A),$o,P,S3) ) ) ) ).

% finite_induct_select
tff(fact_1387_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xa: A,B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),aa(set(A),set(A),insert(A,Xa),B4))
    <=> $ite(
          aa(set(A),$o,member(A,Xa),B4),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4),
          $ite(aa(set(A),$o,member(A,Xa),A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),B4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1388_set__replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xa: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Nb),Xa)) = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_1389_set__replicate__conv__if,axiom,
    ! [A: $tType,Nb: nat,Xa: A] :
      aa(list(A),set(A),set2(A),replicate(A,Nb,Xa)) = $ite(Nb = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_1390_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,minus_minus(nat,Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,minus_minus(nat,Xaa),aa(nat,nat,suc,Xa)))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_1391_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),Xa)) ) ) ) ).

% More_Word.word_l_diffs(2)
tff(fact_1392_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(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),Xa)),Z) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_1393_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),W) ) ) ) ).

% word_diff_ls'(2)
tff(fact_1394_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(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),Xa)) ) ) ) ).

% word_diff_ls'(1)
tff(fact_1395_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(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),Xa)),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa)) ) ) ) ).

% word_l_diffs'(2)
tff(fact_1396_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(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),Xa)),Z) ) ) ) ).

% word_l_diffs'(1)
tff(fact_1397_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),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),Xa)),Xa)) ) ) ) ).

% word_diff_ls''(2)
tff(fact_1398_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),one_one(word(A)))
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_1399_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),zero_zero(word(A))) ) ).

% word_not_simps(1)
tff(fact_1400_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),zero_zero(word(A))) ) ).

% word_coorder.extremum_strict
tff(fact_1401_word__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
        <=> ( zero_zero(word(A)) != Y ) ) ) ).

% word_gt_0
tff(fact_1402_word__neq__0__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),W) ) ) ).

% word_neq_0_conv
tff(fact_1403_word__gt__a__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),Nb)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Nb) ) ) ).

% word_gt_a_gt_0
tff(fact_1404_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),A2)
        <=> ( A2 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_1405_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
        <=> ( Xa != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_1406_word__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa) ) ) ).

% word_not_le
tff(fact_1407_word__le__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),A2)
        <=> ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2) ) ) ).

% word_le_not_less
tff(fact_1408_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_1409_word__gr0__conv__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_1410_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))))
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_1411_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A)))),Xa) ) ) ).

% gt0_iff_gem1
tff(fact_1412_constraint__expand,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Lower: word(A),Upper: word(A)] :
          ( aa(set(word(A)),$o,member(word(A),Xa),aa(fun(word(A),$o),set(word(A)),collect(word(A)),aa(word(A),fun(word(A),$o),aTP_Lamp_ch(word(A),fun(word(A),fun(word(A),$o)),Lower),Upper)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Lower),Xa)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Upper) ) ) ) ).

% constraint_expand
tff(fact_1413_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_1414_word__random,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),X6: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),X6))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),X6)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),Xa)) ) ) ) ).

% word_random
tff(fact_1415_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A),D: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),A2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),D) ) ) ) ) ) ).

% word_sub_mono2
tff(fact_1416_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Va2),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_plus_mcs_3
tff(fact_1417_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Va2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Va2),W) ) ) ) ).

% word_plus_mcs_4
tff(fact_1418_word__add__le__iff2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),J),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff2
tff(fact_1419_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Va2: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Va2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),W))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Va2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Va2),W) ) ) ) ).

% word_plus_mcs_4'
tff(fact_1420_word__add__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),W)),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),W))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),Xa) ) ) ) ).

% word_add_increasing
tff(fact_1421_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Z),Xa)) ) ) ) ).

% word_plus_mono_left
tff(fact_1422_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( ( Xa != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_1423_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_1424_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)) ) ) ) ).

% word_le_plus
tff(fact_1425_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Nb)
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Nb),one_one(word(A))) != zero_zero(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),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_1426_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Nb),one_one(word(A))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Nb) ) ) ).

% plus_one_helper
tff(fact_1427_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),Nb)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),one_one(word(A)))) ) ) ) ).

% word_le_plus_1
tff(fact_1428_word__le__minus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Xa)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)),aa(word(A),word(A),minus_minus(word(A),Xa),Z)) ) ) ) ) ).

% word_le_minus_mono_right
tff(fact_1429_word__le__imp__diff__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Nb)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),M)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Nb),K)),M) ) ) ) ).

% word_le_imp_diff_le
tff(fact_1430_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A)))),Xa) ) ) ).

% word_sub_1_le
tff(fact_1431_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),Xa)
           => ( Nb = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_1432_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( ( Xa != Y )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ) ).

% le_step_down_word_2
tff(fact_1433_le__step__down__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),Nb)
         => ( ( I != Nb )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A)))) ) ) ) ).

% le_step_down_word
tff(fact_1434_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ).

% word_le_minus_one_leq
tff(fact_1435_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A)))),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y) ) ) ).

% word_minus_one_le_leq
tff(fact_1436_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_leq_minus_one_le
tff(fact_1437_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( ( Xa != zero_zero(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A)))),Y) ) ) ) ).

% word_leq_le_minus_one
tff(fact_1438_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa) ) ) ) ).

% le_m1_iff_lt
tff(fact_1439_less__1__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A)))),M)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),M)
            & ( Nb != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_1440_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Nb)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Nb),M)
           => aa(word(A),$o,aa(word(A),fun(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_1441_word__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Nb)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),M)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),M),Nb)),M) ) ) ) ) ).

% word_diff_less
tff(fact_1442_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N3: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),N3),Nb)
         => ( ( N3 != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),minus_minus(word(A),Nb),N3)),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_1443_More__Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),Xa)) ) ) ) ).

% More_Word.word_l_diffs(4)
tff(fact_1444_More__Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% More_Word.word_l_diffs(3)
tff(fact_1445_word__diff__ls_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),W) ) ) ) ).

% word_diff_ls'(4)
tff(fact_1446_word__diff__ls_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_diff_ls'(3)
tff(fact_1447_word__l__diffs_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa)) ) ) ) ).

% word_l_diffs'(4)
tff(fact_1448_word__l__diffs_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)),aa(word(A),word(A),minus_minus(word(A),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% word_l_diffs'(3)
tff(fact_1449_word__diff__ls_H_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),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),Xa)),Xa)) ) ) ) ).

% word_diff_ls''(4)
tff(fact_1450_word__diff__ls_H_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_diff_ls''(3)
tff(fact_1451_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),K)) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_1452_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xa)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(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),Xa)) ) ) ) ).

% word_diff_ls''(1)
tff(fact_1453_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( aa(set(nat),$o,member(nat,I),I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1454_the__elem__eq,axiom,
    ! [A: $tType,Xa: A] : the_elem(A,aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = Xa ).

% the_elem_eq
tff(fact_1455_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ci(set(A),fun(fun(A,B),fun(A,$o)),I5),F))),F4)
           => ( groups1027152243600224163dd_sum(A,B,F,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I),I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F,I5)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_1456_udvd__incr2__K,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),A2: word(A),S2: word(A),K4: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),S2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),S2))
           => ( udvd(A,K4,S2)
             => ( udvd(A,K4,aa(word(A),word(A),minus_minus(word(A),P3),A2))
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),P3)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K4)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4))
                      & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),S2)) ) ) ) ) ) ) ) ) ).

% udvd_incr2_K
tff(fact_1457_is__singletonI,axiom,
    ! [A: $tType,Xa: A] : is_singleton(A,aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_1458_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ci(set(A),fun(fun(A,B),fun(A,$o)),I5),F)))
         => ( groups1027152243600224163dd_sum(A,B,F,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I),I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F,I5)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I5)) ) ) ) ).

% sum_diff1'
tff(fact_1459_set__removeAll,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,Xa),Xs)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_1460_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1962203154675924110t_prod(A,B,P3,aa(set(A),set(A),insert(A,I),I5)) = $ite(aa(set(A),$o,member(A,I),I5),groups1962203154675924110t_prod(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P3,I)),groups1962203154675924110t_prod(A,B,P3,I5))) ) ) ) ).

% prod.insert'
tff(fact_1461_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_1462_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A)] : groups1962203154675924110t_prod(B,A,P3,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_1463_listI__assn__finite,axiom,
    ! [B: $tType,A: $tType,I5: set(nat),A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),I5)
     => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = bot_bot(assn) ) ) ).

% listI_assn_finite
tff(fact_1464_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bg(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1027152243600224163dd_sum(A,B,P3,aa(set(A),set(A),insert(A,I),I5)) = $ite(aa(set(A),$o,member(A,I),I5),groups1027152243600224163dd_sum(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,I)),groups1027152243600224163dd_sum(A,B,P3,I5))) ) ) ) ).

% sum.insert'
tff(fact_1465_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_cj(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_1466_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1962203154675924110t_prod(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ck(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1962203154675924110t_prod(B,A,G,I5) ) ).

% prod.non_neutral'
tff(fact_1467_is__singleton__the__elem,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( A3 = aa(set(A),set(A),insert(A,the_elem(A,A3)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_1468_is__singletonI_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X: A,Y3: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => ( aa(set(A),$o,member(A,Y3),A3)
             => ( X = Y3 ) ) )
       => is_singleton(A,A3) ) ) ).

% is_singletonI'
tff(fact_1469_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cl(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_1470_length__removeAll__less__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,Xa),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_removeAll_less_eq
tff(fact_1471_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cm(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_1472_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T2) = groups1027152243600224163dd_sum(A,B,H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_1473_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,H,I2) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( aa(set(A),$o,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,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_1474_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T2) = groups1027152243600224163dd_sum(A,B,G,S3) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_1475_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S3) = groups1027152243600224163dd_sum(A,B,G,T2) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_1476_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),A3: fun(A,fun(B,assn)),A9: fun(A,fun(B,assn)),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B)] :
      ( ( I5 = I6 )
     => ( ( A3 = A9 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
           => ( ! [I2: nat] :
                  ( aa(set(nat),$o,member(nat,I2),I5)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
                   => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                     => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs4),I2) )
                        & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1477_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B),A3: fun(A,fun(B,assn)),A9: fun(A,fun(B,assn))] :
      ( ( I5 = I6 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
         => ( ! [I2: nat] :
                ( aa(set(nat),$o,member(nat,I2),I5)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
                 => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                   => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs4),I2) )
                      & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) )
                      & ( aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Xsi),I2)) = aa(B,assn,aa(A,fun(B,assn),A9,aa(nat,A,nth(A,Xs4),I2)),aa(nat,B,nth(B,Xsi2),I2)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1478_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,S3) = groups1962203154675924110t_prod(A,B,G,T2) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_1479_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,T2) = groups1962203154675924110t_prod(A,B,G,S3) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_1480_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,H,I2) = one_one(B) ) )
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( groups1962203154675924110t_prod(A,B,G,S3) = groups1962203154675924110t_prod(A,B,H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_1481_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( groups1962203154675924110t_prod(A,B,G,T2) = groups1962203154675924110t_prod(A,B,H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_1482_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bg(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bg(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cl(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_1483_length__removeAll__less,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,Xa),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_1484_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cm(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_1485_udvd__minus__le_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xy: word(A),K: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xy),K)
         => ( udvd(A,Z,Xy)
           => ( udvd(A,Z,K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xy),aa(word(A),word(A),minus_minus(word(A),K),Z)) ) ) ) ) ).

% udvd_minus_le'
tff(fact_1486_is__singleton__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ? [X3: A] : A3 = aa(set(A),set(A),insert(A,X3),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_1487_is__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
     => ~ ! [X: A] : A3 != aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_1488_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ~ aa(set(nat),$o,member(nat,I),I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),insert(nat,I),I5),A3,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1489_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F4: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(set(nat),$o,member(nat,I),I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)),F4),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_1490_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F4: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(set(nat),$o,member(nat,I),I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1491_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( aa(set(list(A)),$o,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_1492_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_1493_Id__on__set,axiom,
    ! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_cn(A,product_prod(A,A)),Xs)) ).

% Id_on_set
tff(fact_1494_bijective__Empty,axiom,
    ! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).

% bijective_Empty
tff(fact_1495_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)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_co(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_1496_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = foldr(nat,nat,times_times(nat),map(list(A),nat,size_size(list(A)),Xss),one_one(nat)) ).

% length_product_lists
tff(fact_1497_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_1498_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_1499_Id__on__empty,axiom,
    ! [A: $tType] : id_on(A,bot_bot(set(A))) = bot_bot(set(product_prod(A,A))) ).

% Id_on_empty
tff(fact_1500_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_1501_ent__iffI,axiom,
    ! [A3: assn,B4: assn] :
      ( entails(A3,B4)
     => ( entails(B4,A3)
       => ( A3 = B4 ) ) ) ).

% ent_iffI
tff(fact_1502_ent__refl,axiom,
    ! [P: assn] : entails(P,P) ).

% ent_refl
tff(fact_1503_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(P,Q)
     => ( entails(Q,R)
       => entails(P,R) ) ) ).

% ent_trans
tff(fact_1504_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,Q) ) ).

% is_entails
tff(fact_1505_fr__rot,axiom,
    ! [A3: assn,B4: assn,C3: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B4),C3)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),A3),C3) ) ).

% fr_rot
tff(fact_1506_fr__refl,axiom,
    ! [A3: assn,B4: assn,C3: assn] :
      ( entails(A3,B4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),C3)) ) ).

% fr_refl
tff(fact_1507_fr__rot__rhs,axiom,
    ! [A3: assn,B4: assn,C3: assn] :
      ( entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),C3))
     => entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),B4)) ) ).

% fr_rot_rhs
tff(fact_1508_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q2: assn] :
      ( entails(P,P2)
     => ( entails(Q,Q2)
       => 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),Q2)) ) ) ).

% ent_star_mono
tff(fact_1509_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F4: assn,Q: assn] :
      ( entails(P,R)
     => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F4))
       => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),F4),Q)
         => entails(Ps,Q) ) ) ) ).

% ent_frame_fwd
tff(fact_1510_cons__rule,axiom,
    ! [A: $tType,P: assn,P2: assn,Q: fun(A,assn),Q2: fun(A,assn),C2: heap_Time_Heap(A)] :
      ( entails(P,P2)
     => ( ! [X: A] : entails(aa(A,assn,Q,X),aa(A,assn,Q2,X))
       => ( hoare_hoare_triple(A,P2,C2,Q)
         => hoare_hoare_triple(A,P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_1511_cons__post__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Q2: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( ! [X: A] : entails(aa(A,assn,Q,X),aa(A,assn,Q2,X))
       => hoare_hoare_triple(A,P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_1512_ent__false,axiom,
    ! [P: assn] : entails(bot_bot(assn),P) ).

% ent_false
tff(fact_1513_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_1514_fi__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ps: assn,F4: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F4))
       => hoare_hoare_triple(A,Ps,C2,aa(assn,fun(A,assn),aTP_Lamp_aa(fun(A,assn),fun(assn,fun(A,assn)),Q),F4)) ) ) ).

% fi_rule
tff(fact_1515_htt__cons__rule,axiom,
    ! [A: $tType,P2: assn,C2: heap_Time_Heap(A),Q2: fun(A,assn),T3: nat,P: assn,Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,P2,C2,Q2,T3)
     => ( entails(P,P2)
       => ( ! [X: A] : entails(aa(A,assn,Q2,X),aa(A,assn,Q,X))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),T3),Ta)
           => time_htt(A,P,C2,Q,Ta) ) ) ) ) ).

% htt_cons_rule
tff(fact_1516_distinct__finite__set,axiom,
    ! [A: $tType,Xa: set(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_cp(set(A),fun(list(A),$o),Xa))) ).

% distinct_finite_set
tff(fact_1517_strict__sorted__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less(A),L)
        <=> ( sorted_wrt(A,ord_less_eq(A),L)
            & distinct(A,L) ) ) ) ).

% strict_sorted_iff
tff(fact_1518_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I3 != J3 )
               => ( aa(nat,A,nth(A,Xs),I3) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_1519_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(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_1520_distinct__length__le,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( distinct(A,Ys)
     => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% distinct_length_le
tff(fact_1521_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( distinct(A,Ys)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_1522_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ? [X: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X) = A3 )
              & sorted_wrt(A,ord_less_eq(A),X)
              & distinct(A,X)
              & ! [Y4: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y4) = A3 )
                    & sorted_wrt(A,ord_less_eq(A),Y4)
                    & distinct(A,Y4) )
                 => ( Y4 = X ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_1523_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
       => ? [X: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y4),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_1524_distinct__idx,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),L: list(B),I: nat,J: nat] :
      ( distinct(A,map(B,A,F,L))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
         => ( ( aa(B,A,F,aa(nat,B,nth(B,L),I)) = aa(B,A,F,aa(nat,B,nth(B,L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_1525_distinct__finite__subset,axiom,
    ! [A: $tType,Xa: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Xa)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_cq(set(A),fun(list(A),$o),Xa))) ) ).

% distinct_finite_subset
tff(fact_1526_distinct__sorted__strict__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_1527_distinct__sorted__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( distinct(A,L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J)) ) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_1528_distinct__sorted__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_1529_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_1530_rule__at__index,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),F4: assn,I: nat,C2: heap_Time_Heap(C),Q2: fun(C,assn),F6: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi)),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F4),C2,Q2)
         => ( ! [R3: C] : entails(aa(C,assn,Q2,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))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),aa(C,assn,F6,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_cr(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),A3),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_1531_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,$o,aa(nat,fun(nat,$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))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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_1532_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,$o,aa(nat,fun(nat,$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))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Rest),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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_1533_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,$o,aa(nat,fun(nat,$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))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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_1534_mergesort__remdups__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( distinct(A,mergesort_remdups(A,L))
          & sorted_wrt(A,ord_less_eq(A),mergesort_remdups(A,L))
          & ( aa(list(A),set(A),set2(A),mergesort_remdups(A,L)) = aa(list(A),set(A),set2(A),L) ) ) ) ).

% mergesort_remdups_correct
tff(fact_1535_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,$o,aa(nat,fun(nat,$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))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y)))) ) ).

% local.ext
tff(fact_1536_listI__assn__reinsert__upd_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xa: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F4: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xa),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(set(nat),$o,member(nat,I),I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,list_update(A,Xs,I,Xa),list_update(B,Xsi,I,Xi))),F4),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_1537_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I,Xa)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_1538_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I),set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_1539_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_1540_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_1541_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_1542_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,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_1543_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),I)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Nb) ) ) ) ) ).

% ivl_subset
tff(fact_1544_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,Nb: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),Nb)
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,Nb)) = set_or7035219750837199246ssThan(A,Nb,M) ) ) ) ).

% ivl_diff
tff(fact_1545_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( list_update(A,Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_1546_listI__assn__wrap__insert,axiom,
    ! [A: $tType,P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set(nat),I: nat,Xs: list(vEBT_VEBT),Xsi: list(vEBT_VEBTi),F4: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Uu,Uua)),Xi)),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Xs))
       => ( aa(set(nat),$o,member(nat,I),I5)
         => ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,I5,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,Xs,I,vEBT_vebt_insert(Uu,Uua)),list_update(vEBT_VEBTi,Xsi,I,Xi))),F4),C2,Q)
           => hoare_hoare_triple(A,P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_1547_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),Xa: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,P3),Xa)),aa(list(A),assn,snga_assn(A,P3),Y)) = bot_bot(assn) ) ).

% snga_same_false
tff(fact_1548_tcd,axiom,
    ! [A: $tType,I: nat,TreeList: list(vEBT_VEBT),TreeList3: list(A),Y: vEBT_VEBT,Xa: vEBT_VEBTi,X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(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)),TreeList3) )
       => 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),Xa)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xa)))),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))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y),list_update(vEBT_VEBTi,Tree_is,I,Xa))),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,Xa))),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,Xa)))) ) ) ).

% tcd
tff(fact_1549_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(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,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_1550_nth__update__invalid,axiom,
    ! [A: $tType,I: nat,L: list(A),J: nat,Xa: A] :
      ( ~ aa(nat,$o,aa(nat,fun(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,Xa)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_update_invalid
tff(fact_1551_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),insert(nat,M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_1552_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),Xaa: vEBT_VEBT,L: nat,Xa: vEBT_VEBTi,Xb: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$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),Xa)),pure_assn(Xb = 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,Xa))),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))),aa(set(nat),set(nat),insert(nat,H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),pure_assn(Xb = 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,Xa)))) ) ) ).

% big_assn_simp'
tff(fact_1553_big__assn__simp,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),L: nat,Xa: vEBT_VEBTi,Xaa: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)),Xa)),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,Xa))),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))),aa(set(nat),set(nat),insert(nat,H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xa))),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,Xa)))) ) ).

% big_assn_simp
tff(fact_1554_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(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_1555_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(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_1556_snga__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(list(A),array(A),aTP_Lamp_cs(list(A),fun(array(A),assn))) ) ).

% snga_prec
tff(fact_1557_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,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D)
             => ( B2 = D ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_1558_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,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_1559_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(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_1560_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),Xa: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
     => ( aa(set(A),$o,member(A,Xa),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xa))),A3) ) ) ).

% set_update_subsetI
tff(fact_1561_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_1562_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_1563_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Nb)
         => aa(nat,$o,P,M5) )
    <=> ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X3) ) ) ).

% all_nat_less_eq
tff(fact_1564_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Nb)
          & aa(nat,$o,P,M5) )
    <=> ? [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X3) ) ) ).

% ex_nat_less_eq
tff(fact_1565_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_1566_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N7),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite2(nat),N7) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_1567_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: A] : aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),list_update(A,L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_1568_in__set__upd__cases,axiom,
    ! [A: $tType,Xa: A,L: list(A),I: nat,Y: A] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
         => ( Xa != Y ) )
       => aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),L)) ) ) ).

% in_set_upd_cases
tff(fact_1569_in__set__upd__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xa = Y )
          | ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),L))
            & ! [Y5: A] : aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),list_update(A,L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_1570_set__update__memI,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xa))) ) ).

% set_update_memI
tff(fact_1571_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xa))),aa(set(A),set(A),insert(A,Xa),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_1572_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xa) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_1573_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(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,Xa)),J) = $ite(I = J,Xa,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_1574_nth__list__update_H,axiom,
    ! [A: $tType,L: list(A),I: nat,Xa: A,J: nat] :
      aa(nat,A,nth(A,list_update(A,L,I,Xa)),J) = $ite(
        ( ( I = J )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)) ),
        Xa,
        aa(nat,A,nth(A,L),J) ) ).

% nth_list_update'
tff(fact_1575_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X2: B] :
      ( ~ aa(set(nat),$o,member(nat,I),I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A3,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X2)) = vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_1576_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_1577_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F: fun(A,B),Xa: A] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,F,aa(nat,A,nth(A,L),I)) = aa(A,B,F,Xa) ) )
     => ( map(A,B,F,list_update(A,L,I,Xa)) = map(A,B,F,L) ) ) ).

% map_upd_eq
tff(fact_1578_atLeastLessThanSuc,axiom,
    ! [M: nat,Nb: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb),aa(set(nat),set(nat),insert(nat,Nb),set_or7035219750837199246ssThan(nat,M,Nb)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_1579_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_1580_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_1581_insert__swap__set__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),set(A),insert(A,aa(nat,A,nth(A,L),I)),aa(list(A),set(A),set2(A),list_update(A,L,I,Xa))) = aa(set(A),set(A),insert(A,Xa),aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_1582_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X2: B] :
      ( ~ aa(set(nat),$o,member(nat,I),I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),insert(nat,I),I5),A3,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,X1),X2)),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1583_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B),F4: 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)),F4) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi)),F4) ) ) ).

% listI_assn_conv'
tff(fact_1584_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ aa(set(A),$o,member(A,A2),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),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_1585_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xa: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(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,Xa)) = aa(set(A),set(A),insert(A,Xa),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,aa(nat,A,nth(A,Xs),Nb)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_1586_listI__assn__reinsert__upd,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xa: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F4: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xa),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(set(nat),$o,member(nat,I),I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,list_update(A,Xs,I,Xa),list_update(B,Xsi,I,Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_1587_nth__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A)] :
          ( aa(nat,$o,aa(nat,fun(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_ct(nat,fun(list(A),fun(array(A),fun(A,assn))),I),Xs),A2)) ) ) ).

% nth_rule
tff(fact_1588_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_cu(array(A),fun(list(A),fun(list(A),assn)),A2),Xs)) ) ).

% freeze_rule
tff(fact_1589_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(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,Xa,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_cv(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I),Xs),A2),Xa)) ) ) ).

% upd_rule
tff(fact_1590_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) )
            & aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),I5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ))),
        finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_cw(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),A3),Xs),Xsi),one_one(assn),I5)) ).

% listI_assn_def
tff(fact_1591_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_cx(array(A),fun(list(A),fun(nat,assn)),A2),Xs)) ) ).

% length_rule
tff(fact_1592_map__distinct__upd__conv,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F: fun(A,B),Xa: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( distinct(A,L)
       => ( list_update(B,map(A,B,F,L),I,Xa) = map(A,B,fun_upd(A,B,F,aa(nat,A,nth(A,L),I),Xa),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_1593_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A2: A,Xs: list(B)] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_cy(fun(B,A),fun(A,fun(B,fun(A,A))),F),A2),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_1594_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,Xa,Xs)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_1595_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,real),Y: real] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F,X)) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),foldr(real,real,plus_plus(real),map(A,real,F,Xs),Y)) ) ).

% listsum_bound
tff(fact_1596_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,real),C2: real,G: fun(A,real),D: real] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F,X)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X))) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),map(A,real,F,Xs),D)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),foldr(real,real,plus_plus(real),map(A,real,G,Xs),zero_zero(real)))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_1597_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,Xa: A,X4: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_am(A,option(B)),Xa,none(B)),X4) = none(B) ).

% empty_upd_none
tff(fact_1598_fold__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(B,fun(A,A)),Z: A] : finite_fold(B,A,F,Z,bot_bot(set(B))) = Z ).

% fold_empty
tff(fact_1599_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),M,A2) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M,A2,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),insert(A,B2),ran(B,A,M)) ) ) ).

% ran_map_upd
tff(fact_1600_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),A2: B,B2: A,Xa: B,Y: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),M,A2,aa(A,option(A),some(A),B2)),Xa) = aa(A,option(A),some(A),Y) )
    <=> ( ( ( Xa = A2 )
          & ( B2 = Y ) )
        | ( ( Xa != A2 )
          & ( aa(B,option(A),M,Xa) = aa(A,option(A),some(A),Y) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_1601_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,Ta: fun(B,option(A)),K: B,Xa: A] :
      ( ( aa(B,option(A),Ta,K) = aa(A,option(A),some(A),Xa) )
     => ( fun_upd(B,option(A),Ta,K,aa(A,option(A),some(A),Xa)) = Ta ) ) ).

% map_upd_triv
tff(fact_1602_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A2: A,Xa: B,Nb: fun(A,option(B)),Y: B] :
      ( ( fun_upd(A,option(B),M,A2,aa(B,option(B),some(B),Xa)) = fun_upd(A,option(B),Nb,A2,aa(B,option(B),some(B),Y)) )
     => ( Xa = Y ) ) ).

% map_upd_eqD1
tff(fact_1603_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C2: B,P: fun(fun(A,B),$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_cz(fun(A,B),fun(B,fun(A,$o)),F),C2)))
     => ( aa(fun(A,B),$o,P,aTP_Lamp_da(B,fun(A,B),C2))
       => ( ! [A4: A,B3: B,F2: fun(A,B)] :
              ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_db(B,fun(fun(A,B),fun(A,$o)),C2),F2)))
             => ( ( aa(A,B,F2,A4) = C2 )
               => ( ( B3 != C2 )
                 => ( aa(fun(A,B),$o,P,F2)
                   => aa(fun(A,B),$o,P,fun_upd(A,B,F2,A4,B3)) ) ) ) )
         => aa(fun(A,B),$o,P,F) ) ) ) ).

% finite_update_induct
tff(fact_1604_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,Ta: fun(A,option(B)),K: A,Xa: B] :
      ~ ! [X: A] : aa(A,option(B),fun_upd(A,option(B),Ta,K,aa(B,option(B),some(B),Xa)),X) = none(B) ).

% map_upd_nonempty
tff(fact_1605_word__to__1__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),one_one(word(A))) = aa(set(word(A)),set(word(A)),insert(word(A),zero_zero(word(A))),bot_bot(set(word(A)))) ) ) ).

% word_to_1_set
tff(fact_1606_set__remove1__subset,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,Xa,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_remove1_subset
tff(fact_1607_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A2,Xs)) ) ) ).

% sorted_remove1
tff(fact_1608_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F,remove1(B,Xa,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_1609_length__remove1,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),remove1(A,Xa,Xs)) = $ite(aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs)),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_remove1
tff(fact_1610_real__nat__list,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),foldr(nat,nat,plus_plus(nat),map(A,nat,F,Xs),C2)) = foldr(real,real,plus_plus(real),map(A,real,aTP_Lamp_dc(fun(A,nat),fun(A,real),F),Xs),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% real_nat_list
tff(fact_1611_new__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xa: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,Nb,Xa),aa(A,fun(array(A),assn),aTP_Lamp_dd(nat,fun(A,fun(array(A),assn)),Nb),Xa)) ) ).

% new_rule
tff(fact_1612_aux,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A2: A,As: 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_de(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),P),A2),As),C2),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),As)))) = 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_cw(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),P),As),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),As)))) ).

% aux
tff(fact_1613_of__list__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : hoare_hoare_triple(array(A),one_one(assn),array_of_list(A,Xs),aa(list(A),fun(array(A),assn),aTP_Lamp_cs(list(A),fun(array(A),assn)),Xs)) ) ).

% of_list_rule
tff(fact_1614_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),Xa: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),A3)),S3)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( finite_fold(A,B,F,Z,aa(set(A),set(A),insert(A,Xa),A3)) = aa(B,B,aa(A,fun(B,B),F,Xa),finite_fold(A,B,F,Z,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_1615_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),A3: set(A),Xa: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S3)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,Xa),A3)
           => ( finite_fold(A,B,F,Z,A3) = aa(B,B,aa(A,fun(B,B),F,Xa),finite_fold(A,B,F,Z,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_1616_subset__mset_Osum__list__update,axiom,
    ! [A: $tType,K: nat,Xs: list(multiset(A)),Xa: multiset(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Xs))
     => ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),list_update(multiset(A),Xs,K,Xa)) = 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)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs)),Xa)),aa(nat,multiset(A),nth(multiset(A),Xs),K)) ) ) ).

% subset_mset.sum_list_update
tff(fact_1617_distinct__foldl__invar,axiom,
    ! [B: $tType,A: $tType,S3: list(A),I5: fun(set(A),fun(B,$o)),Sigma_0: B,F: fun(B,fun(A,B))] :
      ( distinct(A,S3)
     => ( aa(B,$o,aa(set(A),fun(B,$o),I5,aa(list(A),set(A),set2(A),S3)),Sigma_0)
       => ( ! [X: A,It: set(A),Sigma: B] :
              ( aa(set(A),$o,member(A,X),It)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S3))
               => ( aa(B,$o,aa(set(A),fun(B,$o),I5,It),Sigma)
                 => aa(B,$o,aa(set(A),fun(B,$o),I5,aa(set(A),set(A),minus_minus(set(A),It),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F,Sigma),X)) ) ) )
         => aa(B,$o,aa(set(A),fun(B,$o),I5,bot_bot(set(A))),foldl(B,A,F,Sigma_0,S3)) ) ) ) ).

% distinct_foldl_invar
tff(fact_1618_foldr0,axiom,
    ! [Xs: list(real),C2: real,D: real] : foldr(real,real,plus_plus(real),Xs,aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),foldr(real,real,plus_plus(real),Xs,D)),C2) ).

% foldr0
tff(fact_1619_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_1620_foldr__same,axiom,
    ! [Xs: list(real),Y: real] :
      ( ! [X: real,Y3: real] :
          ( aa(set(real),$o,member(real,X),aa(list(real),set(real),set2(real),Xs))
         => ( aa(set(real),$o,member(real,Y3),aa(list(real),set(real),set2(real),Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: real] :
            ( aa(set(real),$o,member(real,X),aa(list(real),set(real),set2(real),Xs))
           => ( X = Y ) )
       => ( foldr(real,real,plus_plus(real),Xs,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_1621_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,real),Bound: real,I: real] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F,X)),Bound) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),map(A,real,F,Xs),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_1622_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_1623_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_1624_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_1625_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(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,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% of_nat_less_iff
tff(fact_1626_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% of_nat_le_iff
tff(fact_1627_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_1628_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_1629_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_1630_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_1631_nth__Cons__Suc,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% nth_Cons_Suc
tff(fact_1632_length__nth__simps_I4_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% length_nth_simps(4)
tff(fact_1633_nth__Cons__0,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),zero_zero(nat)) = Xa ).

% nth_Cons_0
tff(fact_1634_length__nth__simps_I3_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),zero_zero(nat)) = Xa ).

% length_nth_simps(3)
tff(fact_1635_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_1636_foldl__length,axiom,
    ! [A: $tType,L: list(A)] : foldl(nat,A,aTP_Lamp_df(nat,fun(A,nat)),zero_zero(nat),L) = aa(list(A),nat,size_size(list(A)),L) ).

% foldl_length
tff(fact_1637_subset__mset_Osum__list__eq__0__iff,axiom,
    ! [A: $tType,Ns: list(multiset(A))] :
      ( ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns) = zero_zero(multiset(A)) )
    <=> ! [X3: multiset(A)] :
          ( aa(set(multiset(A)),$o,member(multiset(A),X3),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Ns))
         => ( X3 = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.sum_list_eq_0_iff
tff(fact_1638_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_1639_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_1640_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A2: A,Xa: B,Xs: list(B)] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),aa(list(B),list(B),cons(B,Xa),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_1641_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] : enumerate(A,Nb,aa(list(A),list(A),cons(A,Xa),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),cons(product_prod(nat,A),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Nb),Xa)),enumerate(A,aa(nat,nat,suc,Nb),Xs)) ).

% enumerate_simps(2)
tff(fact_1642_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% of_nat_0_less_iff
tff(fact_1643_nth__Cons__pos,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_1644_complete__real,axiom,
    ! [S3: set(real)] :
      ( ? [X4: real] : aa(set(real),$o,member(real,X4),S3)
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa(set(real),$o,member(real,X),S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z4) )
       => ? [Y3: real] :
            ( ! [X4: real] :
                ( aa(set(real),$o,member(real,X4),S3)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Y3) )
            & ! [Z4: real] :
                ( ! [X: real] :
                    ( aa(set(real),$o,member(real,X),S3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),Z4) ) ) ) ) ).

% complete_real
tff(fact_1645_less__eq__real__def,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_real_def
tff(fact_1646_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ! [Y4: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xa)) ) ).

% reals_Archimedean3
tff(fact_1647_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),Xss)) = concat(list(A),map(A,list(list(A)),aTP_Lamp_dg(list(list(A)),fun(A,list(list(A))),Xss),Xs)) ).

% product_lists.simps(2)
tff(fact_1648_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_1649_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_1650_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xa: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xa)) ) ).

% mult_of_nat_commute
tff(fact_1651_nat__less__real__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Nb)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_1652_nat__le__real__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
    <=> aa(real,$o,aa(real,fun(real,$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_1653_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G: fun(A,fun(B,A)),A2: A,F: fun(C,B),Xs: list(C)] : foldl(A,B,G,A2,map(C,B,F,Xs)) = foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_dh(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G),F),A2,Xs) ).

% foldl_map
tff(fact_1654_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_0_le_iff
tff(fact_1655_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_1656_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_1657_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(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,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% of_nat_less_imp_less
tff(fact_1658_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
         => aa(A,$o,aa(A,fun(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_1659_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_1660_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_1661_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),Xa)),C2) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1662_length__nth__simps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,Xa),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_nth_simps(2)
tff(fact_1663_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_1664_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_1665_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))
     => ( Xs != aa(list(A),list(A),cons(A,Xa),Ys) ) ) ).

% impossible_Cons
tff(fact_1666_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xa),Xs))) ).

% set_subset_Cons
tff(fact_1667_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xa),aa(list(A),list(A),cons(A,Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Y),Zs)) ) ) ) ).

% sorted2
tff(fact_1668_list__update__code_I3_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),cons(A,Xa),Xs),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),cons(A,Xa),list_update(A,Xs,I,Y)) ).

% list_update_code(3)
tff(fact_1669_list__update__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),cons(A,Xa),Xs),zero_zero(nat),Y) = aa(list(A),list(A),cons(A,Y),Xs) ).

% list_update_code(2)
tff(fact_1670_replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xa: A] : replicate(A,aa(nat,nat,suc,Nb),Xa) = aa(list(A),list(A),cons(A,Xa),replicate(A,Nb,Xa)) ).

% replicate_Suc
tff(fact_1671_foldl__absorb1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Zs: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),Xa),foldl(A,A,times_times(A),one_one(A),Zs)) = foldl(A,A,times_times(A),Xa,Zs) ) ).

% foldl_absorb1
tff(fact_1672_list__assn_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),A2: A,As: 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),As)),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,As),Cs)) ).

% list_assn.simps(2)
tff(fact_1673_list__assn__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),A2: A,As: 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),As)),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,As),Cs)) ).

% list_assn_simps(2)
tff(fact_1674_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] : aa(A,$o,aa(A,fun(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)),Xa)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1675_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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K) ) ) ).

% of_nat_gt_0
tff(fact_1676_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),Nb)) = aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ) ).

% of_nat_diff
tff(fact_1677_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X3: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,X3),Ys3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_1678_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xa),Ys))
        <=> ( ! [X3: A] :
                ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X3) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_1679_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),cons(A,Xa),Ys))
        <=> ( ! [X3: A] :
                ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X3) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_1680_TBOUND__of__list,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : time_TBOUND(array(A),array_of_list(A,Xs),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% TBOUND_of_list
tff(fact_1681_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),G: fun(A,fun(B,B)),A3: set(A),S2: B,Ta: B,B4: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( finite4664212375090638736ute_on(A,B,S3,G)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => ( aa(A,fun(B,B),F,X) = aa(A,fun(B,B),G,X) ) )
             => ( ( S2 = Ta )
               => ( ( A3 = B4 )
                 => ( finite_fold(A,B,F,S2,A3) = finite_fold(A,B,G,Ta,B4) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_1682_foldl__length__aux,axiom,
    ! [A: $tType,A2: nat,L: list(A)] : foldl(nat,A,aTP_Lamp_df(nat,fun(A,nat)),A2,L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(A),nat,size_size(list(A)),L)) ).

% foldl_length_aux
tff(fact_1683_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_1684_nth__Cons_H,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Nb) = $ite(Nb = zero_zero(nat),Xa,aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ).

% nth_Cons'
tff(fact_1685_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),map(list(A),list(list(A)),aTP_Lamp_dj(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,Nb,Xs))) ).

% n_lists.simps(2)
tff(fact_1686_Id__on__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( id_on(A,A3) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_dk(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A3) ) ) ).

% Id_on_fold
tff(fact_1687_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X21: A,X22: list(A)] : size_list(A,Xa,aa(list(A),list(A),cons(A,X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X21)),size_list(A,Xa,X22))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_1688_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),Xa: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),A3)),S3)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,Xa),A3)
           => ( finite_fold(A,B,F,Z,aa(set(A),set(A),insert(A,Xa),A3)) = aa(B,B,aa(A,fun(B,B),F,Xa),finite_fold(A,B,F,Z,A3)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_1689_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),Xa: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),A3)),S3)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,Xa),A3)
           => ( finite_fold(A,B,F,Z,aa(set(A),set(A),insert(A,Xa),A3)) = finite_fold(A,B,F,aa(B,B,aa(A,fun(B,B),F,Xa),Z),A3) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_1690_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),Xa: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),A3)),S3)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,B,aa(A,fun(B,B),F,Xa),finite_fold(A,B,F,Z,A3)) = finite_fold(A,B,F,aa(B,B,aa(A,fun(B,B),F,Xa),Z),A3) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_1691_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xa != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_1692_nth__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Nb: nat] :
      ( ~ aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Nb) = Xa )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1693_Cons__replicate__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( aa(list(A),list(A),cons(A,Xa),Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xa = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
        & ( Xs = replicate(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xa) ) ) ) ).

% Cons_replicate_eq
tff(fact_1694_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,Xa: A,Xs: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),End) ),
        aa(list(A),list(A),cons(A,Xa),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_1695_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_1696_cnt__cnt__eq,axiom,
    ! [Ta: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Ta)) ).

% cnt_cnt_eq
tff(fact_1697_cnt__non__neg,axiom,
    ! [Ta: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta)) ).

% cnt_non_neg
tff(fact_1698_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa))
    <=> ( Xa = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1699_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_dl(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_1700_length__Cons,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,Xa),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_1701_subset__mset_Oelem__le__sum__list,axiom,
    ! [A: $tType,K: nat,Ns: list(multiset(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Ns))
     => subseteq_mset(A,aa(nat,multiset(A),nth(multiset(A),Ns),K),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns)) ) ).

% subset_mset.elem_le_sum_list
tff(fact_1702_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_dm(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_1703_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(real) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = 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))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList2),zero_zero(real))) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_1704_subset__mset_Oorder__refl,axiom,
    ! [A: $tType,Xa: multiset(A)] : subseteq_mset(A,Xa,Xa) ).

% subset_mset.order_refl
tff(fact_1705_subset__mset_Odual__order_Orefl,axiom,
    ! [A: $tType,A2: multiset(A)] : subseteq_mset(A,A2,A2) ).

% subset_mset.dual_order.refl
tff(fact_1706_subset__mset_Ole__zero__eq,axiom,
    ! [A: $tType,Nb: multiset(A)] :
      ( subseteq_mset(A,Nb,zero_zero(multiset(A)))
    <=> ( Nb = zero_zero(multiset(A)) ) ) ).

% subset_mset.le_zero_eq
tff(fact_1707_subset__mset_Oextremum__unique,axiom,
    ! [A: $tType,A2: multiset(A)] :
      ( subseteq_mset(A,A2,zero_zero(multiset(A)))
    <=> ( A2 = zero_zero(multiset(A)) ) ) ).

% subset_mset.extremum_unique
tff(fact_1708_subset__mset_Oadd__le__cancel__left,axiom,
    ! [A: $tType,C2: multiset(A),A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2))
    <=> subseteq_mset(A,A2,B2) ) ).

% subset_mset.add_le_cancel_left
tff(fact_1709_subset__mset_Oadd__le__cancel__right,axiom,
    ! [A: $tType,A2: multiset(A),C2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2))
    <=> subseteq_mset(A,A2,B2) ) ).

% subset_mset.add_le_cancel_right
tff(fact_1710_mset__subset__eq__mono__add__left__cancel,axiom,
    ! [A: $tType,C3: multiset(A),A3: multiset(A),B4: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),B4))
    <=> subseteq_mset(A,A3,B4) ) ).

% mset_subset_eq_mono_add_left_cancel
tff(fact_1711_mset__subset__eq__mono__add__right__cancel,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B4: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),C3))
    <=> subseteq_mset(A,A3,B4) ) ).

% mset_subset_eq_mono_add_right_cancel
tff(fact_1712_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_1713_subset__mset_Oadd__le__same__cancel1,axiom,
    ! [A: $tType,B2: multiset(A),A2: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A2),B2)
    <=> subseteq_mset(A,A2,zero_zero(multiset(A))) ) ).

% subset_mset.add_le_same_cancel1
tff(fact_1714_subset__mset_Oadd__le__same__cancel2,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),B2),B2)
    <=> subseteq_mset(A,A2,zero_zero(multiset(A))) ) ).

% subset_mset.add_le_same_cancel2
tff(fact_1715_subset__mset_Ole__add__same__cancel1,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),B2))
    <=> subseteq_mset(A,zero_zero(multiset(A)),B2) ) ).

% subset_mset.le_add_same_cancel1
tff(fact_1716_subset__mset_Ole__add__same__cancel2,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A2))
    <=> subseteq_mset(A,zero_zero(multiset(A)),B2) ) ).

% subset_mset.le_add_same_cancel2
tff(fact_1717_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_1718_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_1719_subset__mset_Odiff__add,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2)),A2) = B2 ) ) ).

% subset_mset.diff_add
tff(fact_1720_subset__mset_Oadd__diff__assoc,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2)) = 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)),C2),B2)),A2) ) ) ).

% subset_mset.add_diff_assoc
tff(fact_1721_subset__mset_Oadd__diff__assoc2,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2)),C2) = 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)),B2),C2)),A2) ) ) ).

% subset_mset.add_diff_assoc2
tff(fact_1722_mset__subset__eq__multiset__union__diff__commute,axiom,
    ! [A: $tType,B4: multiset(A),A3: multiset(A),C3: multiset(A)] :
      ( subseteq_mset(A,B4,A3)
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),A3),B4)),C3) = 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),C3)),B4) ) ) ).

% mset_subset_eq_multiset_union_diff_commute
tff(fact_1723_subset__mset_Ofinite__has__minimal,axiom,
    ! [A: $tType,A3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
     => ( ( A3 != bot_bot(set(multiset(A))) )
       => ? [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),A3)
            & ! [Xa2: multiset(A)] :
                ( aa(set(multiset(A)),$o,member(multiset(A),Xa2),A3)
               => ( subseteq_mset(A,Xa2,X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% subset_mset.finite_has_minimal
tff(fact_1724_subset__mset_Ofinite__has__maximal,axiom,
    ! [A: $tType,A3: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
     => ( ( A3 != bot_bot(set(multiset(A))) )
       => ? [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),A3)
            & ! [Xa2: multiset(A)] :
                ( aa(set(multiset(A)),$o,member(multiset(A),Xa2),A3)
               => ( subseteq_mset(A,X,Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% subset_mset.finite_has_maximal
tff(fact_1725_pochhammer__of__nat,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: nat,Nb: nat] : comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),Xa),Nb) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,Xa,Nb)) ) ).

% pochhammer_of_nat
tff(fact_1726_subset__mset_Otrans,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( subseteq_mset(A,B2,C2)
       => subseteq_mset(A,A2,C2) ) ) ).

% subset_mset.trans
tff(fact_1727_subset__mset_Oeq__iff,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( ( A2 = B2 )
    <=> ( subseteq_mset(A,A2,B2)
        & subseteq_mset(A,B2,A2) ) ) ).

% subset_mset.eq_iff
tff(fact_1728_subset__mset_Oantisym,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( subseteq_mset(A,B2,A2)
       => ( A2 = B2 ) ) ) ).

% subset_mset.antisym
tff(fact_1729_subset__mset_Oeq__refl,axiom,
    ! [A: $tType,Xa: multiset(A),Y: multiset(A)] :
      ( ( Xa = Y )
     => subseteq_mset(A,Xa,Y) ) ).

% subset_mset.eq_refl
tff(fact_1730_subset__mset_Oorder__trans,axiom,
    ! [A: $tType,Xa: multiset(A),Y: multiset(A),Z: multiset(A)] :
      ( subseteq_mset(A,Xa,Y)
     => ( subseteq_mset(A,Y,Z)
       => subseteq_mset(A,Xa,Z) ) ) ).

% subset_mset.order_trans
tff(fact_1731_subset__mset_Oantisym__conv,axiom,
    ! [A: $tType,Y: multiset(A),Xa: multiset(A)] :
      ( subseteq_mset(A,Y,Xa)
     => ( subseteq_mset(A,Xa,Y)
      <=> ( Xa = Y ) ) ) ).

% subset_mset.antisym_conv
tff(fact_1732_subset__mset_Oorder__eq__iff,axiom,
    ! [A: $tType,Xa: multiset(A),Y: multiset(A)] :
      ( ( Xa = Y )
    <=> ( subseteq_mset(A,Xa,Y)
        & subseteq_mset(A,Y,Xa) ) ) ).

% subset_mset.order_eq_iff
tff(fact_1733_subset__mset_Oorder__antisym,axiom,
    ! [A: $tType,Xa: multiset(A),Y: multiset(A)] :
      ( subseteq_mset(A,Xa,Y)
     => ( subseteq_mset(A,Y,Xa)
       => ( Xa = Y ) ) ) ).

% subset_mset.order_antisym
tff(fact_1734_subset__mset_Oord__eq__le__trans,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( ( A2 = B2 )
     => ( subseteq_mset(A,B2,C2)
       => subseteq_mset(A,A2,C2) ) ) ).

% subset_mset.ord_eq_le_trans
tff(fact_1735_subset__mset_Oord__le__eq__trans,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( ( B2 = C2 )
       => subseteq_mset(A,A2,C2) ) ) ).

% subset_mset.ord_le_eq_trans
tff(fact_1736_subset__mset_Odual__order_Otrans,axiom,
    ! [A: $tType,B2: multiset(A),A2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,B2,A2)
     => ( subseteq_mset(A,C2,B2)
       => subseteq_mset(A,C2,A2) ) ) ).

% subset_mset.dual_order.trans
tff(fact_1737_subset__mset_Odual__order_Oeq__iff,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( ( A2 = B2 )
    <=> ( subseteq_mset(A,B2,A2)
        & subseteq_mset(A,A2,B2) ) ) ).

% subset_mset.dual_order.eq_iff
tff(fact_1738_subset__mset_Odual__order_Oantisym,axiom,
    ! [A: $tType,B2: multiset(A),A2: multiset(A)] :
      ( subseteq_mset(A,B2,A2)
     => ( subseteq_mset(A,A2,B2)
       => ( A2 = B2 ) ) ) ).

% subset_mset.dual_order.antisym
tff(fact_1739_subset__mset_Ofinite__has__minimal2,axiom,
    ! [A: $tType,A3: set(multiset(A)),A2: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
     => ( aa(set(multiset(A)),$o,member(multiset(A),A2),A3)
       => ? [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),A3)
            & subseteq_mset(A,X,A2)
            & ! [Xa2: multiset(A)] :
                ( aa(set(multiset(A)),$o,member(multiset(A),Xa2),A3)
               => ( subseteq_mset(A,Xa2,X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% subset_mset.finite_has_minimal2
tff(fact_1740_subset__mset_Ofinite__has__maximal2,axiom,
    ! [A: $tType,A3: set(multiset(A)),A2: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
     => ( aa(set(multiset(A)),$o,member(multiset(A),A2),A3)
       => ? [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),A3)
            & subseteq_mset(A,A2,X)
            & ! [Xa2: multiset(A)] :
                ( aa(set(multiset(A)),$o,member(multiset(A),Xa2),A3)
               => ( subseteq_mset(A,X,Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% subset_mset.finite_has_maximal2
tff(fact_1741_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1742_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_1743_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_1744_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_1745_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$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_1746_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_1747_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(3)
tff(fact_1748_zle__int,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% zle_int
tff(fact_1749_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,aa(int,fun(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_1750_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_1751_subset__mset_Ozero__le,axiom,
    ! [A: $tType,Xa: multiset(A)] : subseteq_mset(A,zero_zero(multiset(A)),Xa) ).

% subset_mset.zero_le
tff(fact_1752_subset__mset_Obot__least,axiom,
    ! [A: $tType,A2: multiset(A)] : subseteq_mset(A,zero_zero(multiset(A)),A2) ).

% subset_mset.bot_least
tff(fact_1753_subset__mset_Oextremum__uniqueI,axiom,
    ! [A: $tType,A2: multiset(A)] :
      ( subseteq_mset(A,A2,zero_zero(multiset(A)))
     => ( A2 = zero_zero(multiset(A)) ) ) ).

% subset_mset.extremum_uniqueI
tff(fact_1754_empty__le,axiom,
    ! [A: $tType,A3: multiset(A)] : subseteq_mset(A,zero_zero(multiset(A)),A3) ).

% empty_le
tff(fact_1755_subset__mset_Oadd__mono,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A),D: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( subseteq_mset(A,C2,D)
       => subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),D)) ) ) ).

% subset_mset.add_mono
tff(fact_1756_subset__mset_Oless__eqE,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ~ ! [C5: multiset(A)] : B2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C5) ) ).

% subset_mset.less_eqE
tff(fact_1757_subset__mset_Ole__iff__add,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
    <=> ? [C6: multiset(A)] : B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C6) ) ).

% subset_mset.le_iff_add
tff(fact_1758_subset__mset_Oadd__left__mono,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2)) ) ).

% subset_mset.add_left_mono
tff(fact_1759_subset__mset_Oadd__right__mono,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2)) ) ).

% subset_mset.add_right_mono
tff(fact_1760_subset__mset_Oadd__le__imp__le__left,axiom,
    ! [A: $tType,C2: multiset(A),A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2))
     => subseteq_mset(A,A2,B2) ) ).

% subset_mset.add_le_imp_le_left
tff(fact_1761_subset__mset_Oadd__le__imp__le__right,axiom,
    ! [A: $tType,A2: multiset(A),C2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2))
     => subseteq_mset(A,A2,B2) ) ).

% subset_mset.add_le_imp_le_right
tff(fact_1762_mset__subset__eq__add__left,axiom,
    ! [A: $tType,A3: multiset(A),B4: multiset(A)] : subseteq_mset(A,A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B4)) ).

% mset_subset_eq_add_left
tff(fact_1763_mset__subset__eq__mono__add,axiom,
    ! [A: $tType,A3: multiset(A),B4: multiset(A),C3: multiset(A),D4: multiset(A)] :
      ( subseteq_mset(A,A3,B4)
     => ( subseteq_mset(A,C3,D4)
       => subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),D4)) ) ) ).

% mset_subset_eq_mono_add
tff(fact_1764_mset__subset__eq__add__right,axiom,
    ! [A: $tType,B4: multiset(A),A3: multiset(A)] : subseteq_mset(A,B4,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B4)) ).

% mset_subset_eq_add_right
tff(fact_1765_mset__subset__eq__exists__conv,axiom,
    ! [A: $tType,A3: multiset(A),B4: multiset(A)] :
      ( subseteq_mset(A,A3,B4)
    <=> ? [C7: multiset(A)] : B4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C7) ) ).

% mset_subset_eq_exists_conv
tff(fact_1766_diff__subset__eq__self,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A)] : subseteq_mset(A,aa(multiset(A),multiset(A),minus_minus(multiset(A),M6),N7),M6) ).

% diff_subset_eq_self
tff(fact_1767_nat__less__as__int,axiom,
    ! [X4: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_less_as_int
tff(fact_1768_nat__leq__as__int,axiom,
    ! [X4: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_leq_as_int
tff(fact_1769_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_1770_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_1771_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$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_1772_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$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_1773_subset__mset_Olift__Suc__antimono__le,axiom,
    ! [A: $tType,F: fun(nat,multiset(A)),Nb: nat,N3: nat] :
      ( ! [N: nat] : subseteq_mset(A,aa(nat,multiset(A),F,aa(nat,nat,suc,N)),aa(nat,multiset(A),F,N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
       => subseteq_mset(A,aa(nat,multiset(A),F,N3),aa(nat,multiset(A),F,Nb)) ) ) ).

% subset_mset.lift_Suc_antimono_le
tff(fact_1774_subset__mset_Olift__Suc__mono__le,axiom,
    ! [A: $tType,F: fun(nat,multiset(A)),Nb: nat,N3: nat] :
      ( ! [N: nat] : subseteq_mset(A,aa(nat,multiset(A),F,N),aa(nat,multiset(A),F,aa(nat,nat,suc,N)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
       => subseteq_mset(A,aa(nat,multiset(A),F,Nb),aa(nat,multiset(A),F,N3)) ) ) ).

% subset_mset.lift_Suc_mono_le
tff(fact_1775_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_1776_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Nb)) ) ) ).

% pochhammer_pos
tff(fact_1777_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1778_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => ( comm_s3205402744901411588hammer(A,A2,Nb) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1779_word__le__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ? [N: nat] : Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),N)) ) ) ).

% word_le_add
tff(fact_1780_subset__mset_Oadd__decreasing,axiom,
    ! [A: $tType,A2: multiset(A),C2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,zero_zero(multiset(A)))
     => ( subseteq_mset(A,C2,B2)
       => subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2),B2) ) ) ).

% subset_mset.add_decreasing
tff(fact_1781_subset__mset_Oadd__increasing,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,zero_zero(multiset(A)),A2)
     => ( subseteq_mset(A,B2,C2)
       => subseteq_mset(A,B2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2)) ) ) ).

% subset_mset.add_increasing
tff(fact_1782_subset__mset_Oadd__decreasing2,axiom,
    ! [A: $tType,C2: multiset(A),A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,C2,zero_zero(multiset(A)))
     => ( subseteq_mset(A,A2,B2)
       => subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2),B2) ) ) ).

% subset_mset.add_decreasing2
tff(fact_1783_subset__mset_Oadd__increasing2,axiom,
    ! [A: $tType,C2: multiset(A),B2: multiset(A),A2: multiset(A)] :
      ( subseteq_mset(A,zero_zero(multiset(A)),C2)
     => ( subseteq_mset(A,B2,A2)
       => subseteq_mset(A,B2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C2)) ) ) ).

% subset_mset.add_increasing2
tff(fact_1784_subset__mset_Oadd__nonneg__nonneg,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,zero_zero(multiset(A)),A2)
     => ( subseteq_mset(A,zero_zero(multiset(A)),B2)
       => subseteq_mset(A,zero_zero(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),B2)) ) ) ).

% subset_mset.add_nonneg_nonneg
tff(fact_1785_subset__mset_Oadd__nonpos__nonpos,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,zero_zero(multiset(A)))
     => ( subseteq_mset(A,B2,zero_zero(multiset(A)))
       => subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),B2),zero_zero(multiset(A))) ) ) ).

% subset_mset.add_nonpos_nonpos
tff(fact_1786_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
    ! [A: $tType,Xa: multiset(A),Y: multiset(A)] :
      ( subseteq_mset(A,zero_zero(multiset(A)),Xa)
     => ( subseteq_mset(A,zero_zero(multiset(A)),Y)
       => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa),Y) = zero_zero(multiset(A)) )
        <=> ( ( Xa = zero_zero(multiset(A)) )
            & ( Y = zero_zero(multiset(A)) ) ) ) ) ) ).

% subset_mset.add_nonneg_eq_0_iff
tff(fact_1787_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
    ! [A: $tType,Xa: multiset(A),Y: multiset(A)] :
      ( subseteq_mset(A,Xa,zero_zero(multiset(A)))
     => ( subseteq_mset(A,Y,zero_zero(multiset(A)))
       => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa),Y) = zero_zero(multiset(A)) )
        <=> ( ( Xa = zero_zero(multiset(A)) )
            & ( Y = zero_zero(multiset(A)) ) ) ) ) ) ).

% subset_mset.add_nonpos_eq_0_iff
tff(fact_1788_Diff__eq__empty__iff__mset,axiom,
    ! [A: $tType,A3: multiset(A),B4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),A3),B4) = zero_zero(multiset(A)) )
    <=> subseteq_mset(A,A3,B4) ) ).

% Diff_eq_empty_iff_mset
tff(fact_1789_subset__mset_Ole__add__diff,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => subseteq_mset(A,C2,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)),B2),C2)),A2)) ) ).

% subset_mset.le_add_diff
tff(fact_1790_subset__mset_Ole__diff__conv2,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( subseteq_mset(A,C2,aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2))
      <=> subseteq_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A2),B2) ) ) ).

% subset_mset.le_diff_conv2
tff(fact_1791_subset__mset_Odiff__add__assoc,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( 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)),C2),B2)),A2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2)) ) ) ).

% subset_mset.diff_add_assoc
tff(fact_1792_subset__mset_Odiff__add__assoc2,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( 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)),B2),C2)),A2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2)),C2) ) ) ).

% subset_mset.diff_add_assoc2
tff(fact_1793_subset__mset_Odiff__diff__right,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C2),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2)) = 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)),C2),A2)),B2) ) ) ).

% subset_mset.diff_diff_right
tff(fact_1794_subset__mset_Oadd__diff__inverse,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2)) = B2 ) ) ).

% subset_mset.add_diff_inverse
tff(fact_1795_subset__mset_Ole__imp__diff__is__add,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( subseteq_mset(A,A2,B2)
     => ( subseteq_mset(A,A2,B2)
       => ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),A2) = C2 )
        <=> ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A2) ) ) ) ) ).

% subset_mset.le_imp_diff_is_add
tff(fact_1796_subset__eq__diff__conv,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B4: multiset(A)] :
      ( subseteq_mset(A,aa(multiset(A),multiset(A),minus_minus(multiset(A),A3),C3),B4)
    <=> subseteq_mset(A,A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),C3)) ) ).

% subset_eq_diff_conv
tff(fact_1797_multiset__diff__union__assoc,axiom,
    ! [A: $tType,C3: multiset(A),B4: multiset(A),A3: multiset(A)] :
      ( subseteq_mset(A,C3,B4)
     => ( 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)),C3) = 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),C3)) ) ) ).

% multiset_diff_union_assoc
tff(fact_1798_size__mset__mono,axiom,
    ! [A: $tType,A3: multiset(A),B4: multiset(A)] :
      ( subseteq_mset(A,A3,B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),A3)),aa(multiset(A),nat,size_size(multiset(A)),B4)) ) ).

% size_mset_mono
tff(fact_1799_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_1800_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Nb)) ) ) ).

% pochhammer_nonneg
tff(fact_1801_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_1802_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xa: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Xa),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
         => aa(int,$o,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Xa)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_1803_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_1804_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_1805_size__Diff__submset,axiom,
    ! [A: $tType,M6: multiset(A),M9: multiset(A)] :
      ( subseteq_mset(A,M6,M9)
     => ( aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),M9),M6)) = aa(nat,nat,minus_minus(nat,aa(multiset(A),nat,size_size(multiset(A)),M9)),aa(multiset(A),nat,size_size(multiset(A)),M6)) ) ) ).

% size_Diff_submset
tff(fact_1806_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType,Xs: list(multiset(A))] :
      ( ! [X: multiset(A)] :
          ( aa(set(multiset(A)),$o,member(multiset(A),X),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs))
         => subseteq_mset(A,zero_zero(multiset(A)),X) )
     => ( ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs) = zero_zero(multiset(A)) )
      <=> ! [X3: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X3),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs))
           => ( X3 = zero_zero(multiset(A)) ) ) ) ) ).

% subset_mset.sum_list_nonneg_eq_0_iff
tff(fact_1807_subset__mset_Omember__le__sum__list,axiom,
    ! [A: $tType,Xa: multiset(A),Xs: list(multiset(A))] :
      ( aa(set(multiset(A)),$o,member(multiset(A),Xa),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs))
     => subseteq_mset(A,Xa,groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs)) ) ).

% subset_mset.member_le_sum_list
tff(fact_1808_subset__mset_Osum__list__nonpos,axiom,
    ! [A: $tType,Xs: list(multiset(A))] :
      ( ! [X: multiset(A)] :
          ( aa(set(multiset(A)),$o,member(multiset(A),X),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs))
         => subseteq_mset(A,X,zero_zero(multiset(A))) )
     => subseteq_mset(A,groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs),zero_zero(multiset(A))) ) ).

% subset_mset.sum_list_nonpos
tff(fact_1809_subset__mset_Osum__list__nonneg,axiom,
    ! [A: $tType,Xs: list(multiset(A))] :
      ( ! [X: multiset(A)] :
          ( aa(set(multiset(A)),$o,member(multiset(A),X),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs))
         => subseteq_mset(A,zero_zero(multiset(A)),X) )
     => subseteq_mset(A,zero_zero(multiset(A)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs)) ) ).

% subset_mset.sum_list_nonneg
tff(fact_1810_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_1811_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_1812_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_1813_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_1814_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Xa: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A)))),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2) ) ) ) ).

% word_1_0
tff(fact_1815_subset__Collect__iff,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),A3),P)))
      <=> ! [X3: A] :
            ( aa(set(A),$o,member(A,X3),B4)
           => aa(A,$o,P,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_1816_subset__CollectI,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),Q: fun(A,$o),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),B4)
           => ( aa(A,$o,Q,X)
             => aa(A,$o,P,X) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),B4),Q))),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% subset_CollectI
tff(fact_1817_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Nb: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( comm_s3205402744901411588hammer(A,Z,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,minus_minus(nat,Nb),M))) ) ) ) ).

% pochhammer_product
tff(fact_1818_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))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList),zero_zero(real))) ).

% VEBT_internal.cnt.simps(2)
tff(fact_1819_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(real) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = 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))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList2),zero_zero(real))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_1820_Tbuildupi__buildupi_H,axiom,
    ! [Nb: nat] : aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Nb)) = vEBT_V9176841429113362141ildupi(Nb) ).

% Tbuildupi_buildupi'
tff(fact_1821_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B3: real,C5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A4),B3)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B3),C5)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),C5)
                 => aa(real,$o,aa(real,fun(real,$o),P,A4),C5) ) ) ) )
       => ( ! [X: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
               => ? [D3: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
                    & ! [A4: real,B3: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),X)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B3)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,minus_minus(real,B3),A4)),D3) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A4),B3) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_1822_Set__filter__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( filter3(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_dn(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).

% Set_filter_fold
tff(fact_1823_subset__mset_Osum__mono2,axiom,
    ! [B: $tType,A: $tType,B4: set(A),A3: set(A),F: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => ( ! [B3: A] :
              ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),minus_minus(set(A),B4),A3))
             => subseteq_mset(B,zero_zero(multiset(B)),aa(A,multiset(B),F,B3)) )
         => subseteq_mset(B,groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,A3),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,B4)) ) ) ) ).

% subset_mset.sum_mono2
tff(fact_1824_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,Va2: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),Va2))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),Va2)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).

% graph_map_upd
tff(fact_1825_word__count__from__top,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] :
          ( ( Nb != zero_zero(word(A)) )
         => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),Nb) = aa(set(word(A)),set(word(A)),aa(set(word(A)),fun(set(word(A)),set(word(A))),sup_sup(set(word(A))),set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A))))),aa(set(word(A)),set(word(A)),insert(word(A),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A)))),bot_bot(set(word(A))))) ) ) ) ).

% word_count_from_top
tff(fact_1826_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_do(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_1827_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_dp(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_1828_sup__Some,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Xa: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),aa(A,option(A),some(A),Xa)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_Some
tff(fact_1829_Un__empty,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> ( ( A3 = bot_bot(set(A)) )
        & ( B4 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_1830_Un__subset__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3) ) ) ).

% Un_subset_iff
tff(fact_1831_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_dq(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_1832_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_dr(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_1833_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_am(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_1834_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ).

% zle_add1_eq_le
tff(fact_1835_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),aa(int,int,minus_minus(int,Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% zle_diff1_eq
tff(fact_1836_subset__mset_Osum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType,F4: set(A),F: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,F4) = zero_zero(multiset(B)) )
      <=> ! [X3: A] :
            ( aa(set(A),$o,member(A,X3),F4)
           => ( aa(A,multiset(B),F,X3) = zero_zero(multiset(B)) ) ) ) ) ).

% subset_mset.sum_eq_0_iff
tff(fact_1837_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z)
    <=> ? [N6: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N6)) ) ).

% zle_iff_zadd
tff(fact_1838_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_1839_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1840_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1841_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_1842_pos__mult__pos__ge,axiom,
    ! [Xa: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),Xa)) ) ) ).

% pos_mult_pos_ge
tff(fact_1843_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1844_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q4) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1845_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q4: int,R2: int,B6: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$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),Q5)),R4)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1846_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q4: int,R2: int,B6: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1847_q__pos__lemma,axiom,
    ! [B6: int,Q5: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1848_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => aa(int,$o,aa(int,fun(int,$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_1849_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_1850_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% add1_zle_eq
tff(fact_1851_incr__mult__lemma,axiom,
    ! [D: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(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,aa(int,fun(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_1852_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1853_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_1854_decr__mult__lemma,axiom,
    ! [D: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(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,aa(int,fun(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_1855_less__1__helper,axiom,
    ! [Nb: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,Nb),one_one(int))),M) ) ).

% less_1_helper
tff(fact_1856_minusinfinity,axiom,
    ! [D: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$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))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P1,X) ) )
         => ( ? [X_1: int] : aa(int,$o,P1,X_1)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_1857_plusinfinity,axiom,
    ! [D: int,P2: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$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))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P2,X) ) )
         => ( ? [X_1: int] : aa(int,$o,P2,X_1)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_1858_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_1859_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_1860_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1861_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$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_1862_less__int__code_I1_J,axiom,
    ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_1863_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_av(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_disj_eq
tff(fact_1864_Un__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ds(set(A),fun(set(A),fun(A,$o)),A3),B4)) ).

% Un_def
tff(fact_1865_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_1866_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_1867_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_1868_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_1869_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_1870_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_1871_imp__le__cong,axiom,
    ! [Xa: int,X6: int,P: $o,P2: $o] :
      ( ( Xa = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_1872_conj__le__cong,axiom,
    ! [Xa: int,X6: int,P: $o,P2: $o] :
      ( ( Xa = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
            & (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_1873_verit__la__generic,axiom,
    ! [A2: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),Xa)
      | ( A2 = Xa )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),A2) ) ).

% verit_la_generic
tff(fact_1874_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_1875_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% boolean_algebra.disj_zero_right
tff(fact_1876_foldl__un__empty__eq,axiom,
    ! [A: $tType,I: set(A),Ww: list(set(A))] : foldl(set(A),set(A),sup_sup(set(A)),I,Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I),foldl(set(A),set(A),sup_sup(set(A)),bot_bot(set(A)),Ww)) ).

% foldl_un_empty_eq
tff(fact_1877_Un__empty__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B4) = B4 ).

% Un_empty_left
tff(fact_1878_Un__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),bot_bot(set(A))) = A3 ).

% Un_empty_right
tff(fact_1879_Un__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B4: set(A),D4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),D4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),D4)) ) ) ).

% Un_mono
tff(fact_1880_Un__least,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)),C3) ) ) ).

% Un_least
tff(fact_1881_Un__upper1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) ).

% Un_upper1
tff(fact_1882_Un__upper2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) ).

% Un_upper2
tff(fact_1883_Un__absorb1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_1884_Un__absorb2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = A3 ) ) ).

% Un_absorb2
tff(fact_1885_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))
     => ~ ! [A10: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),A3)
           => ! [B7: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B7),B4)
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A10),B7) ) ) ) ) ).

% subset_UnE
tff(fact_1886_subset__Un__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_1887_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A3: set(A)] : filter3(A,P,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_dt(fun(A,$o),fun(set(A),fun(A,$o)),P),A3)) ).

% Set.filter_def
tff(fact_1888_insert__def,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : aa(set(A),set(A),insert(A,A2),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ca(A,fun(A,$o),A2))),B4) ).

% insert_def
tff(fact_1889_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V9176841429113362141ildupi(zero_zero(nat)) = one_one(int) ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_1890_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))),A3) ).

% insert_is_Un
tff(fact_1891_Un__singleton__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),Xa: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B4 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) )
          & ( B4 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) )
          & ( B4 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1892_singleton__Un__iff,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B4 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) )
          & ( B4 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) )
          & ( B4 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1893_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_1894_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B4)),C3)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) ) ).

% Diff_subset_conv
tff(fact_1895_Diff__partition,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B4),A3)) = B4 ) ) ).

% Diff_partition
tff(fact_1896_in__graphD,axiom,
    ! [A: $tType,B: $tType,K: A,Va2: B,M: fun(A,option(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),Va2)),graph(A,B,M))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),Va2) ) ) ).

% in_graphD
tff(fact_1897_in__graphI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,Va2: A] :
      ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),Va2) )
     => aa(set(product_prod(B,A)),$o,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Va2)),graph(B,A,M)) ) ).

% in_graphI
tff(fact_1898_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,zero_zero(nat))) = one_one(int) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_1899_subset__mset_Osum__nonpos,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(A,multiset(B))] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => subseteq_mset(B,aa(A,multiset(B),F,X),zero_zero(multiset(B))) )
     => subseteq_mset(B,groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,A3),zero_zero(multiset(B))) ) ).

% subset_mset.sum_nonpos
tff(fact_1900_subset__mset_Osum__nonneg,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(A,multiset(B))] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => subseteq_mset(B,zero_zero(multiset(B)),aa(A,multiset(B),F,X)) )
     => subseteq_mset(B,zero_zero(multiset(B)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,A3)) ) ).

% subset_mset.sum_nonneg
tff(fact_1901_subset__mset_Osum__mono,axiom,
    ! [B: $tType,A: $tType,K4: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
      ( ! [I2: A] :
          ( aa(set(A),$o,member(A,I2),K4)
         => subseteq_mset(B,aa(A,multiset(B),F,I2),aa(A,multiset(B),G,I2)) )
     => subseteq_mset(B,groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,K4),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),G,K4)) ) ).

% subset_mset.sum_mono
tff(fact_1902_subset__mset_Osum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType,S2: set(A),Ta: set(B),G: fun(B,multiset(C)),I: fun(B,A),F: fun(A,multiset(C))] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),Ta)
       => ( ! [X: B] :
              ( aa(set(B),$o,member(B,X),Ta)
             => subseteq_mset(C,zero_zero(multiset(C)),aa(B,multiset(C),G,X)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),S2)
               => ? [Xa2: B] :
                    ( aa(set(B),$o,member(B,Xa2),Ta)
                    & ( aa(B,A,I,Xa2) = X )
                    & subseteq_mset(C,aa(A,multiset(C),F,X),aa(B,multiset(C),G,Xa2)) ) )
           => subseteq_mset(C,groups3894954378712506084id_sum(multiset(C),A,plus_plus(multiset(C)),zero_zero(multiset(C)),F,S2),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),G,Ta)) ) ) ) ) ).

% subset_mset.sum_le_included
tff(fact_1903_subset__mset_Osum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => subseteq_mset(B,zero_zero(multiset(B)),aa(A,multiset(B),F,X)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,A3) = zero_zero(multiset(B)) )
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),A3)
             => ( aa(A,multiset(B),F,X3) = zero_zero(multiset(B)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_eq_0_iff
tff(fact_1904_subset__mset_Osum__nonneg__0,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F: fun(A,multiset(B)),I: A] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),S2)
           => subseteq_mset(B,zero_zero(multiset(B)),aa(A,multiset(B),F,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,S2) = zero_zero(multiset(B)) )
         => ( aa(set(A),$o,member(A,I),S2)
           => ( aa(A,multiset(B),F,I) = zero_zero(multiset(B)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_0
tff(fact_1905_subset__mset_Osum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F: fun(A,multiset(B)),B4: multiset(B),I: A] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),S2)
           => subseteq_mset(B,zero_zero(multiset(B)),aa(A,multiset(B),F,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,S2) = B4 )
         => ( aa(set(A),$o,member(A,I),S2)
           => subseteq_mset(B,aa(A,multiset(B),F,I),B4) ) ) ) ) ).

% subset_mset.sum_nonneg_leq_bound
tff(fact_1906_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),Xa) = Xa ) ).

% sup_bot_left
tff(fact_1907_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% sup_bot_right
tff(fact_1908_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_1909_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = bot_bot(A) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_1910_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = bot_bot(A) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1911_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A2) = A2 ) ).

% sup_bot.left_neutral
tff(fact_1912_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1913_merge__pure__or,axiom,
    ! [A2: $o,B2: $o] :
      aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn((A2))),pure_assn((B2))) = pure_assn(( (A2)
        | (B2) )) ).

% merge_pure_or
tff(fact_1914_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_1915_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_1916_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),bot_bot(A)) = A2 ) ).

% sup_bot.right_neutral
tff(fact_1917_sup__None__1,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),none(A)),Y) = Y ) ).

% sup_None_1
tff(fact_1918_sup__None__2,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Xa: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),Xa),none(A)) = Xa ) ).

% sup_None_2
tff(fact_1919_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B)),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R),S3)) ) ).

% sup_Un_eq2
tff(fact_1920_star__or__dist1,axiom,
    ! [A3: assn,B4: assn,C3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A3),B4)),C3) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C3)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),C3)) ).

% star_or_dist1
tff(fact_1921_star__or__dist2,axiom,
    ! [C3: assn,A3: assn,B4: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A3),B4)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),A3)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),B4)) ).

% star_or_dist2
tff(fact_1922_ent__disjE,axiom,
    ! [A3: assn,C3: assn,B4: assn] :
      ( entails(A3,C3)
     => ( entails(B4,C3)
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A3),B4),C3) ) ) ).

% ent_disjE
tff(fact_1923_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),R)
     => entails(P,R) ) ).

% ent_disjI1
tff(fact_1924_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),R)
     => entails(Q,R) ) ).

% ent_disjI2
tff(fact_1925_ent__disjI1_H,axiom,
    ! [A3: assn,B4: assn,C3: assn] :
      ( entails(A3,B4)
     => entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B4),C3)) ) ).

% ent_disjI1'
tff(fact_1926_ent__disjI2_H,axiom,
    ! [A3: assn,C3: assn,B4: assn] :
      ( entails(A3,C3)
     => entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B4),C3)) ) ).

% ent_disjI2'
tff(fact_1927_ent__disjI1__direct,axiom,
    ! [A3: assn,B4: assn] : entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A3),B4)) ).

% ent_disjI1_direct
tff(fact_1928_ent__disjI2__direct,axiom,
    ! [B4: assn,A3: assn] : entails(B4,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A3),B4)) ).

% ent_disjI2_direct
tff(fact_1929_split__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),R: fun(A,assn),Q: assn] :
      ( hoare_hoare_triple(A,P,C2,R)
     => ( hoare_hoare_triple(A,Q,C2,R)
       => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),C2,R) ) ) ).

% split_rule
tff(fact_1930_norm__assertion__simps_I6_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),bot_bot(assn)) = Xa ).

% norm_assertion_simps(6)
tff(fact_1931_norm__assertion__simps_I5_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),bot_bot(assn)),Xa) = Xa ).

% norm_assertion_simps(5)
tff(fact_1932_sup__Un__eq,axiom,
    ! [A: $tType,R: set(A),S3: set(A),X4: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),S3)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R),S3)) ) ).

% sup_Un_eq
tff(fact_1933_sup__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),B4))) ).

% sup_set_def
tff(fact_1934_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_1935_if__rule__split,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap(A),Q1: fun(A,assn),G: heap_Time_Heap(A),Q22: fun(A,assn),Q: fun(A,assn)] :
      ( ( (B2)
       => hoare_hoare_triple(A,P,F,Q1) )
     => ( ( ~ (B2)
         => hoare_hoare_triple(A,P,G,Q22) )
       => ( ! [X: A] : entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Q1,X)),pure_assn((B2)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Q22,X)),pure_assn(~ (B2)))),aa(A,assn,Q,X))
         => hoare_hoare_triple(A,P,
              $ite((B2),F,G),
              Q) ) ) ) ).

% if_rule_split
tff(fact_1936_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(4)
tff(fact_1937_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(3)
tff(fact_1938_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xa)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa) ) ) ) ).

% le_supE
tff(fact_1939_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xa) ) ) ) ).

% le_supI
tff(fact_1940_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge1
tff(fact_1941_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge2
tff(fact_1942_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI1
tff(fact_1943_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI2
tff(fact_1944_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_1945_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D)) ) ) ) ).

% sup_mono
tff(fact_1946_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),Xa) ) ) ) ).

% sup_least
tff(fact_1947_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_1948_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_1949_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% sup.orderI
tff(fact_1950_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F,X),Y3))
         => ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F,X),Y3))
           => ( ! [X: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,Y3),Z3)),X) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F,Xa),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_1951_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_1952_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_1953_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Xa ) ) ) ).

% sup_absorb1
tff(fact_1954_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_1955_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.boundedE
tff(fact_1956_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2) ) ) ) ).

% sup.boundedI
tff(fact_1957_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.order_iff
tff(fact_1958_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded1
tff(fact_1959_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded2
tff(fact_1960_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb_iff1
tff(fact_1961_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_1962_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_1963_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_1964_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_1965_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_1966_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_1967_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% sup.strict_boundedE
tff(fact_1968_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_1969_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb3
tff(fact_1970_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI2
tff(fact_1971_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI1
tff(fact_1972_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1973_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I2: int,J2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_1974_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),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),Q4)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q4 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1975_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_du(A,fun(A,$o)),aTP_Lamp_dv(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1976_merge__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L1: list(A),L22: list(A)] :
          ( ( distinct(A,L1)
            & sorted_wrt(A,ord_less_eq(A),L1) )
         => ( ( distinct(A,L22)
              & sorted_wrt(A,ord_less_eq(A),L22) )
           => ( distinct(A,merge(A,L1,L22))
              & sorted_wrt(A,ord_less_eq(A),merge(A,L1,L22))
              & ( aa(list(A),set(A),set2(A),merge(A,L1,L22)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),L1)),aa(list(A),set(A),set2(A),L22)) ) ) ) ) ) ).

% merge_correct
tff(fact_1977_time__array__nth,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),I: nat,H: heap_ext(product_unit)] :
          time_time(A,array_nth(A,P3,I),H) = $ite(time_fails(A,array_nth(A,P3,I),H),zero_zero(nat),one_one(nat)) ) ).

% time_array_nth
tff(fact_1978_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_idempotent
tff(fact_1979_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_abs
tff(fact_1980_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_1981_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_1982_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_1983_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_1984_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_1985_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_1986_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),A2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% abs_mult_self_eq
tff(fact_1987_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ).

% abs_of_nat
tff(fact_1988_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_1989_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% abs_le_self_iff
tff(fact_1990_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_1991_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_1992_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_1993_norm__assertion__simps_I15_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y)),Z) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Z)) ).

% norm_assertion_simps(15)
tff(fact_1994_norm__assertion__simps_I32_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Xa) = Xa ).

% norm_assertion_simps(32)
tff(fact_1995_assn__aci_I5_J,axiom,
    ! [Xa: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Xa) ).

% assn_aci(5)
tff(fact_1996_assn__aci_I7_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Z)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Z)) ).

% assn_aci(7)
tff(fact_1997_assn__aci_I8_J,axiom,
    ! [Xa: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),Y) ).

% assn_aci(8)
tff(fact_1998_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_1999_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_self
tff(fact_2000_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% abs_le_D1
tff(fact_2001_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2002_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_2003_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2)) ) ).

% abs_minus_commute
tff(fact_2004_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_2005_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_zero
tff(fact_2006_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2007_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2008_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq
tff(fact_2009_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D)) ) ) ) ).

% abs_mult_less
tff(fact_2010_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2011_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_2012_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_2013_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S3)
    <=> ! [M5: int] :
        ? [N6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M5),aa(int,int,abs_abs(int),N6))
          & aa(set(int),$o,member(int,N6),S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_2014_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xa: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),E) )
         => ( Xa = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2015_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2016_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),Xa) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa)) ) ) ) ).

% abs_mult_pos
tff(fact_2017_merge_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X1: A,L1: list(A),X2: A,L22: list(A)] :
          merge(A,aa(list(A),list(A),cons(A,X1),L1),aa(list(A),list(A),cons(A,X2),L22)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),X1),X2),
            aa(list(A),list(A),cons(A,X1),merge(A,L1,aa(list(A),list(A),cons(A,X2),L22))),
            $ite(X1 = X2,aa(list(A),list(A),cons(A,X1),merge(A,L1,L22)),aa(list(A),list(A),cons(A,X2),merge(A,aa(list(A),list(A),cons(A,X1),L1),L22))) ) ) ).

% merge.simps(3)
tff(fact_2018_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xa),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2019_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq4
tff(fact_2020_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),C2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),D)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2021_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xa),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A2),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2022_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_2023_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),Xa))) ) ).

% abs_add_one_gt_zero
tff(fact_2024_time__refines,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),C4: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( refine_Imp_refines(A,C2,C4)
     => ( ~ time_fails(A,C4,H)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,C2,H)),time_time(A,C4,H)) ) ) ).

% time_refines
tff(fact_2025_nat__intermed__int__val,axiom,
    ! [M: nat,Nb: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,M)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,Nb))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
                & ( aa(nat,int,F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2026_decr__lemma,axiom,
    ! [D: int,Xa: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,Xa),Z))),one_one(int))),D))),Z) ) ).

% decr_lemma
tff(fact_2027_incr__lemma,axiom,
    ! [D: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,Xa),Z))),one_one(int))),D))) ) ).

% incr_lemma
tff(fact_2028_time__array__upd,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xa: A,P3: array(A),H: heap_ext(product_unit)] :
          time_time(array(A),array_upd(A,I,Xa,P3),H) = $ite(time_fails(array(A),array_upd(A,I,Xa,P3),H),zero_zero(nat),one_one(nat)) ) ).

% time_array_upd
tff(fact_2029_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2030_bezw__0,axiom,
    ! [Xa: nat] : bezw(Xa,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_2031_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3) ) )
        | ? [L3: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),zero_zero(int)) )
            & ( L3 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R5) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L3) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L3))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2032_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3) )
               => ( ( sgn_sgn(int,R3) = sgn_sgn(int,A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2033_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))) = remove1(A,Xa,aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_2034_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),Xa: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),A3)),S3)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( finite_fold(A,B,F,Z,aa(set(A),set(A),insert(A,Xa),A3)) = aa(B,B,aa(A,fun(B,B),F,Xa),finite_fold(A,B,F,Z,A3)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_2035_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),Xa: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xa),A3)),S3)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( finite_fold(A,B,F,Z,aa(set(A),set(A),insert(A,Xa),A3)) = finite_fold(A,B,F,aa(B,B,aa(A,fun(B,B),F,Xa),Z),A3) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_2036_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_2037_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_2038_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_2039_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),sgn_sgn(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_2040_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),sgn_sgn(A,A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_2041_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_2042_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_2043_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_2044_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_2045_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_2046_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_2047_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_2048_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),sgn_sgn(A,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_2049_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),sgn_sgn(A,A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_2050_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)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_2051_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xa)),aa(A,A,abs_abs(A),Xa)) = Xa ) ).

% mult_sgn_abs
tff(fact_2052_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_2053_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_2054_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_2055_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_2056_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_2057_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_2058_lemma__interval,axiom,
    ! [A2: real,Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [Y4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xa),Y4))),D2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_2059_sin__bound__lemma,axiom,
    ! [Xa: real,Y: real,U: real,Va2: real] :
      ( ( Xa = Y )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),U)),Va2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),U)),Y))),Va2) ) ) ).

% sin_bound_lemma
tff(fact_2060_lemma__interval__lt,axiom,
    ! [A2: real,Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [Y4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xa),Y4))),D2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2061_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,Y: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xa)),sgn_sgn(A,Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_2062_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),insert(A,Xa),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_2063_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( ( sgn_sgn(A,Xa) = zero_zero(A) )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_2064_sgn__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sgn_sgn(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_2065_zero__le__sgn__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sgn_sgn(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% zero_le_sgn_iff
tff(fact_2066_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F),Xa),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_insort
tff(fact_2067_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,Xa),A3)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),insert(A,Xa),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_2068_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa),Xs)) ) ).

% insort_left_comm
tff(fact_2069_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xa: A,Y: A,Ys: list(A)] :
          aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F),Xa),aa(list(A),list(A),cons(A,Y),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Xa)),aa(A,B,F,Y)),aa(list(A),list(A),cons(A,Xa),aa(list(A),list(A),cons(A,Y),Ys)),aa(list(A),list(A),cons(A,Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F),Xa),Ys))) ) ).

% insort_key.simps(2)
tff(fact_2070_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_2071_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F: fun(A,B),A2: A] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A2)),aa(A,B,F,X)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F),A2),Xs) = aa(list(A),list(A),cons(A,A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_2072_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xa: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),Xa),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs)) ) ) ).

% sorted_insort_key
tff(fact_2073_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_dx(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_2074_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_2075_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,Xa),A3)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_2076_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_dw(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_2077_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),D4: set(A),Xa: A] :
      fun_upd(A,option(B),restrict_map(A,B,M,D4),Xa,none(B)) = $ite(aa(set(A),$o,member(A,Xa),D4),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D4),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),restrict_map(A,B,M,D4)) ).

% fun_upd_None_restrict
tff(fact_2078_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xa: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_dy(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,Xa,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,Xa),C2),aa(nat,nat,minus_minus(nat,Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_2079_total__on__singleton,axiom,
    ! [A: $tType,Xa: A] : total_on(A,aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_2080_nth__step__trancl,axiom,
    ! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),Nb: nat,M: nat] :
      ( ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N))),aa(nat,A,nth(A,Xs),N))),R) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
         => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),Nb)),aa(nat,A,nth(A,Xs),M))),transitive_trancl(A,R)) ) ) ) ).

% nth_step_trancl
tff(fact_2081_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_2082_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K,M)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_2083_prod__encode__eq,axiom,
    ! [Xa: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),nat,nat_prod_encode,Xa) = aa(product_prod(nat,nat),nat,nat_prod_encode,Y) )
    <=> ( Xa = Y ) ) ).

% prod_encode_eq
tff(fact_2084_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_bp(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_2085_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F),A3) = bot_bot(set(A)) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_2086_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image(B,A,F),A3) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_2087_image__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A)] : aa(set(B),set(A),image(B,A,F),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_2088_zip__eq__zip__same__len,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),A6: list(A),B6: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B6) )
       => ( ( zip(A,B,A2,B2) = zip(A,B,A6,B6) )
        <=> ( ( A2 = A6 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_2089_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D4: set(A),X4: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_am(A,option(B)),D4),X4) = none(B) ).

% restrict_map_empty
tff(fact_2090_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S3) = S3 ) ).

% image_add_0
tff(fact_2091_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),X4: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X4) = none(B) ).

% restrict_map_to_empty
tff(fact_2092_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_2093_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_dz(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_2094_trancl__single,axiom,
    ! [A: $tType,A2: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),bot_bot(set(product_prod(A,A)))) ).

% trancl_single
tff(fact_2095_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A,Y: option(B),D4: set(A)] :
      restrict_map(A,B,fun_upd(A,option(B),M,Xa,Y),D4) = $ite(aa(set(A),$o,member(A,Xa),D4),fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D4),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),Xa,Y),restrict_map(A,B,M,D4)) ).

% restrict_fun_upd
tff(fact_2096_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,Xa: A,D4: set(A),M: fun(A,option(B)),Y: option(B)] :
      ( aa(set(A),$o,member(A,Xa),D4)
     => ( fun_upd(A,option(B),restrict_map(A,B,M,D4),Xa,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D4),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),Xa,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_2097_nth__image__indices,axiom,
    ! [A: $tType,L: list(A)] : aa(set(nat),set(A),image(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(A),set(A),set2(A),L) ).

% nth_image_indices
tff(fact_2098_le__map__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M: fun(A,option(B)),X5: set(A)] : aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M,X5)),M) ) ).

% le_map_restrict
tff(fact_2099_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F),A3))
     => ~ ! [X: B] :
            ( ( B2 = aa(B,A,F,X) )
           => ~ aa(set(B),$o,member(B,X),A3) ) ) ).

% imageE
tff(fact_2100_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image(B,A,F),aa(set(C),set(B),image(C,B,G),A3)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ea(fun(B,A),fun(fun(C,B),fun(C,A)),F),G)),A3) ).

% image_image
tff(fact_2101_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A3: set(B),P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_eb(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F),A3),P)) = aa(set(B),set(A),image(B,A,F),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ec(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F),A3),P))) ).

% Compr_image_eq
tff(fact_2102_zip__inj,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),A6: list(A),B6: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B6) )
       => ( ( zip(A,B,A2,B2) = zip(A,B,A6,B6) )
         => ( ( A2 = A6 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
tff(fact_2103_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B8: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B8),aa(set(B),set(A),image(B,A,F),A3))
         => aa(set(A),$o,P,B8) )
    <=> ! [B8: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B8),A3)
         => aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),B8)) ) ) ).

% all_subset_image
tff(fact_2104_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F),A3))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A3)
          & ( B4 = aa(set(B),set(A),image(B,A,F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2105_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A3: set(B),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F),A3)),B4)
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
         => aa(set(A),$o,member(A,aa(B,A,F,X3)),B4) ) ) ).

% image_subset_iff
tff(fact_2106_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F),A3))
     => ~ ! [C8: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),A3)
           => ( B4 != aa(set(B),set(A),image(B,A,F),C8) ) ) ) ).

% subset_imageE
tff(fact_2107_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(A,B),B4: set(B)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(set(B),$o,member(B,aa(A,B,F,X)),B4) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),B4) ) ).

% image_subsetI
tff(fact_2108_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B4)) ) ).

% image_mono
tff(fact_2109_trancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V: set(product_prod(A,A)),Xa: product_prod(A,A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),U2),V)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),Xa),transitive_trancl(A,U2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),Xa),transitive_trancl(A,V)) ) ) ).

% trancl_mono_mp
tff(fact_2110_trancl__sub,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),transitive_trancl(A,R)) ).

% trancl_sub
tff(fact_2111_restrict__map__subset__eq,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),R: set(A),M7: fun(A,option(B)),R6: set(A)] :
      ( ( restrict_map(A,B,M,R) = M7 )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R6),R)
       => ( restrict_map(A,B,M,R6) = restrict_map(A,B,M7,R6) ) ) ) ).

% restrict_map_subset_eq
tff(fact_2112_restrict__map__eq_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: set(B),K: B,Va2: A] :
      ( ( aa(B,option(A),restrict_map(B,A,M,A3),K) = aa(A,option(A),some(A),Va2) )
    <=> ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),Va2) )
        & aa(set(B),$o,member(B,K),A3) ) ) ).

% restrict_map_eq(2)
tff(fact_2113_total__on__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R2) ).

% total_on_empty
tff(fact_2114_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(A,B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image(A,B,F),A3))
       => ? [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
            & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_ed(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A3),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_2115_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F: fun(A,B),B4: set(B)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
         => aa(set(B),$o,member(B,aa(A,B,F,X)),B4) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),aa(fun(A,$o),set(A),collect(A),P))),B4) ) ).

% image_Collect_subsetI
tff(fact_2116_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ aa(set(nat),$o,member(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),A3)) ).

% zero_notin_Suc_image
tff(fact_2117_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B8: set(A)] :
          ( ( aa(set(A),$o,finite_finite2(A),B8)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B8),aa(set(B),set(A),image(B,A,F),A3)) )
         => aa(set(A),$o,P,B8) )
    <=> ! [B8: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),B8)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B8),A3) )
         => aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),B8)) ) ) ).

% all_finite_subset_image
tff(fact_2118_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ? [B8: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B8)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B8),aa(set(B),set(A),image(B,A,F),A3))
          & aa(set(A),$o,P,B8) )
    <=> ? [B8: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),B8)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B8),A3)
          & aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2119_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F),A3))
       => ? [C8: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),A3)
            & aa(set(B),$o,finite_finite2(B),C8)
            & ( B4 = aa(set(B),set(A),image(B,A,F),C8) ) ) ) ) ).

% finite_subset_image
tff(fact_2120_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),F: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),aa(set(A),set(B),image(A,B,F),A3))
       => aa(set(B),$o,finite_finite2(B),B4) ) ) ).

% finite_surj
tff(fact_2121_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),B4: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),aa(set(B),set(A),image(B,A,F),A3)),aa(set(B),set(A),image(B,A,F),B4))),aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),minus_minus(set(B),A3),B4))) ).

% image_diff_subset
tff(fact_2122_trancl__sub__insert__trancl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xa: product_prod(A,A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R)),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),Xa),R))) ).

% trancl_sub_insert_trancl
tff(fact_2123_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F: fun(nat,A),Nb: nat] :
      ( ( A3 = aa(set(nat),set(A),image(nat,A,F),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_as(nat,fun(nat,$o)),Nb))) )
     => aa(set(A),$o,finite_finite2(A),A3) ) ).

% nat_seg_image_imp_finite
tff(fact_2124_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
    <=> ? [N6: nat,F7: fun(nat,A)] : A3 = aa(set(nat),set(A),image(nat,A,F7),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_as(nat,fun(nat,$o)),N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_2125_image__constant,axiom,
    ! [A: $tType,B: $tType,Xa: A,A3: set(A),C2: B] :
      ( aa(set(A),$o,member(A,Xa),A3)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_da(B,fun(A,B),C2)),A3) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_2126_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A3: set(B)] :
      aa(set(B),set(A),image(B,A,aTP_Lamp_bx(A,fun(B,A),C2)),A3) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert(A,C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_2127_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y: A,M: fun(B,option(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,Y),ran(B,A,restrict_map(B,A,M,A3)))
     => ? [X: B] :
          ( aa(set(B),$o,member(B,X),A3)
          & ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) ) ) ) ).

% ran_restrictD
tff(fact_2128_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = map(A,product_prod(A,A),aTP_Lamp_cn(A,product_prod(A,A)),Xs) ).

% zip_same_conv_map
tff(fact_2129_pair__list__split,axiom,
    ! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
      ~ ! [L12: list(A),L23: list(B)] :
          ( ( L = zip(A,B,L12,L23) )
         => ( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L23) )
           => ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L23) ) ) ) ).

% pair_list_split
tff(fact_2130_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(A,B),Xa: A] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( aa(set(A),$o,member(A,Y3),A3)
           => ( aa(A,B,F,Y3) = aa(A,B,F,Xa) ) )
       => ( the_elem(B,aa(set(A),set(B),image(A,B,F),A3)) = aa(A,B,F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_2131_restrict__map__upd,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),S3: set(A),K: A,Va2: B] : fun_upd(A,option(B),restrict_map(A,B,F,S3),K,aa(B,option(B),some(B),Va2)) = restrict_map(A,B,fun_upd(A,option(B),F,K,aa(B,option(B),some(B),Va2)),aa(set(A),set(A),insert(A,K),S3)) ).

% restrict_map_upd
tff(fact_2132_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K: A,Va2: B,M: fun(A,option(B)),A3: set(A)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),Va2)),graph(A,B,restrict_map(A,B,M,A3)))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),Va2) ) ) ).

% graph_restrictD(2)
tff(fact_2133_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_1
tff(fact_2134_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_2
tff(fact_2135_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(B),$o,member(B,Y),aa(list(B),set(B),set2(B),Ys))
       => ~ ! [X: A] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2136_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xa: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
       => ~ ! [Y3: B] : ~ aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2137_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L4: list(A),F: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L4) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,F,aa(nat,A,nth(A,L),I2)) = aa(A,B,F,aa(nat,A,nth(A,L4),I2)) ) )
       => ( aa(set(A),set(B),image(A,B,F),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image(A,B,F),aa(list(A),set(A),set2(A),L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_2138_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xa: B,L: list(B)] :
      ( aa(set(A),$o,member(A,aa(B,A,F,Xa)),aa(set(B),set(A),image(B,A,F),aa(list(B),set(B),set2(B),L)))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),L))
          & ( aa(B,A,F,aa(nat,B,nth(B,L),I3)) = aa(B,A,F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_2139_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_2140_map__zip1,axiom,
    ! [A: $tType,B: $tType,K: B,L: list(A)] : map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ee(B,fun(A,product_prod(A,B))),K),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K)) ).

% map_zip1
tff(fact_2141_map__zip2,axiom,
    ! [A: $tType,B: $tType,K: A,L: list(B)] : map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K),L) ).

% map_zip2
tff(fact_2142_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),D4: set(A),Xa: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M,D4),Xa,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D4),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),Xa,Y) ).

% fun_upd_restrict
tff(fact_2143_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),set(B),image(A,B,F),A3) = finite_fold(A,set(B),aTP_Lamp_ef(fun(A,B),fun(A,fun(set(B),set(B))),F),bot_bot(set(B)),A3) ) ) ).

% image_fold_insert
tff(fact_2144_trancl__empty,axiom,
    ! [A: $tType] : transitive_trancl(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ).

% trancl_empty
tff(fact_2145_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xa: B,Y: A,A3: set(B)] :
      aa(set(B),set(A),image(B,A,fun_upd(B,A,F,Xa,Y)),A3) = $ite(aa(set(B),$o,member(B,Xa),A3),aa(set(A),set(A),insert(A,Y),aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),insert(B,Xa),bot_bot(set(B)))))),aa(set(B),set(A),image(B,A,F),A3)) ).

% fun_upd_image
tff(fact_2146_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),aa(set(A),set(A),minus_minus(set(A),S2),Ta)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),S2)),aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_diff
tff(fact_2147_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D4: set(A),M: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D4)
       => ( restrict_map(A,B,map_upds(A,B,M,Xs,Ys),D4) = map_upds(A,B,restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D4),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_2148_trancl__mono,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P3),transitive_trancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P3),transitive_trancl(A,S2)) ) ) ).

% trancl_mono
tff(fact_2149_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(set(nat),set(A),image(nat,A,nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),take(A,L,Xs)) ) ) ).

% nth_image
tff(fact_2150_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,Xa,aa(B,option(B),some(B),Y)),uminus_uminus(set(A),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))) = restrict_map(A,B,M,uminus_uminus(set(A),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_2151_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus(A,A2) = uminus_uminus(A,B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_2152_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : uminus_uminus(A,uminus_uminus(A,A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_2153_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( uminus_uminus(A,A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_2154_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = uminus_uminus(A,A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_2155_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( uminus_uminus(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_2156_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = uminus_uminus(A,A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_2157_add_Oinverse__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( uminus_uminus(A,zero_zero(A)) = zero_zero(A) ) ) ).

% add.inverse_neutral
tff(fact_2158_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),uminus_uminus(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_2159_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,Xa)),uminus_uminus(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% compl_le_compl_iff
tff(fact_2160_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,B2)),uminus_uminus(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% neg_less_iff_less
tff(fact_2161_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,Xa)),uminus_uminus(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% compl_less_compl_iff
tff(fact_2162_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: 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),uminus_uminus(A,A2)),uminus_uminus(A,B2)) ) ).

% minus_add_distrib
tff(fact_2163_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_2164_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),uminus_uminus(A,A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_2165_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),uminus_uminus(A,B2)) = uminus_uminus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_right
tff(fact_2166_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,A2)),uminus_uminus(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) ) ).

% minus_mult_minus
tff(fact_2167_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,A2)),B2) = uminus_uminus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_left
tff(fact_2168_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: 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_2169_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),uminus_uminus(A,A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus_cancel
tff(fact_2170_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),uminus_uminus(A,A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus
tff(fact_2171_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,uminus_uminus(A,A2)) = uminus_uminus(A,sgn_sgn(A,A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_2172_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),uminus_uminus(set(A),B4)),uminus_uminus(set(A),A3)) ) ).

% Compl_anti_mono
tff(fact_2173_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),uminus_uminus(set(A),A3)),uminus_uminus(set(A),B4))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3) ) ).

% Compl_subset_Compl_iff
tff(fact_2174_image__map__upd,axiom,
    ! [B: $tType,A: $tType,Xa: A,A3: set(A),M: fun(A,option(B)),Y: B] :
      ( ~ aa(set(A),$o,member(A,Xa),A3)
     => ( aa(set(A),set(option(B)),image(A,option(B),fun_upd(A,option(B),M,Xa,aa(B,option(B),some(B),Y))),A3) = aa(set(A),set(option(B)),image(A,option(B),M),A3) ) ) ).

% image_map_upd
tff(fact_2175_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_2176_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),uminus_uminus(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_2177_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_2178_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),uminus_uminus(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_2179_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_0_iff_less
tff(fact_2180_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),uminus_uminus(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_2181_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_pos
tff(fact_2182_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),uminus_uminus(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_2183_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,A2)),A2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_2184_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),uminus_uminus(A,A2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_2185_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = uminus_uminus(A,A2) ) ).

% diff_0
tff(fact_2186_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,minus_minus(A,zero_zero(A)),B2) = uminus_uminus(A,B2) ) ).

% verit_minus_simplify(3)
tff(fact_2187_mult__minus1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,one_one(A))),Z) = uminus_uminus(A,Z) ) ).

% mult_minus1
tff(fact_2188_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),uminus_uminus(A,one_one(A))) = uminus_uminus(A,Z) ) ).

% mult_minus1_right
tff(fact_2189_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),uminus_uminus(A,B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_2190_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),uminus_uminus(A,A2)),B2) = aa(A,A,minus_minus(A,B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_2191_take__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] : take(A,aa(nat,nat,suc,Nb),aa(list(A),list(A),cons(A,Xa),Xs)) = aa(list(A),list(A),cons(A,Xa),take(A,Nb,Xs)) ).

% take_Suc_Cons
tff(fact_2192_take__all,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)
     => ( take(A,Nb,Xs) = Xs ) ) ).

% take_all
tff(fact_2193_take__all__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( take(A,Nb,Xs) = Xs )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% take_all_iff
tff(fact_2194_nth__take,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(nat,A,nth(A,take(A,Nb,Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).

% nth_take
tff(fact_2195_take__update__cancel,axiom,
    ! [A: $tType,Nb: nat,M: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( take(A,Nb,list_update(A,Xs,M,Y)) = take(A,Nb,Xs) ) ) ).

% take_update_cancel
tff(fact_2196_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,one_one(A))),one_one(A)) = zero_zero(A) ) ) ).

% add_neg_numeral_special(8)
tff(fact_2197_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)),uminus_uminus(A,one_one(A))) = zero_zero(A) ) ) ).

% add_neg_numeral_special(7)
tff(fact_2198_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,uminus_uminus(A,one_one(A))),uminus_uminus(A,one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_2199_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = uminus_uminus(A,A2) ) ) ) ).

% abs_of_nonpos
tff(fact_2200_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),uminus_uminus(set(A),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))
    <=> ~ aa(set(A),$o,member(A,B2),A3) ) ).

% subset_Compl_singleton
tff(fact_2201_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,M: fun(A,option(B)),Ys: list(B),Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( map_upds(A,B,M,Xs,list_update(B,Ys,I,Y)) = map_upds(A,B,M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_2202_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A2: A,As: list(A),B2: B,Bs: list(B)] : map_upds(A,B,M,aa(list(A),list(A),cons(A,A2),As),aa(list(B),list(B),cons(B,B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),M,A2,aa(B,option(B),some(B),B2)),As,Bs) ).

% map_upds_Cons
tff(fact_2203_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A2: A,As: list(A),M: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),As))
     => ( map_upds(A,B,fun_upd(A,option(B),M,A2,aa(B,option(B),some(B),B2)),As,Bs) = fun_upd(A,option(B),map_upds(A,B,M,As,Bs),A2,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_2204_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( sgn_sgn(A,A2) = uminus_uminus(A,one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_2205_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus(A,A2) = B2 )
        <=> ( uminus_uminus(A,B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_2206_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = uminus_uminus(A,B2) )
        <=> ( B2 = uminus_uminus(A,A2) ) ) ) ).

% equation_minus_iff
tff(fact_2207_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_eh(fun(A,$o),fun(A,$o),P)) = uminus_uminus(set(A),aa(fun(A,$o),set(A),collect(A),P)) ).

% Collect_neg_eq
tff(fact_2208_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : uminus_uminus(set(A),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ei(set(A),fun(A,$o),A3)) ).

% Compl_eq
tff(fact_2209_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),uminus_uminus(A,A2)) ) ) ).

% le_imp_neg_le
tff(fact_2210_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),A2) ) ) ).

% minus_le_iff
tff(fact_2211_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),uminus_uminus(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),uminus_uminus(A,A2)) ) ) ).

% le_minus_iff
tff(fact_2212_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,Y)),uminus_uminus(A,Xa)) ) ) ).

% compl_mono
tff(fact_2213_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),uminus_uminus(A,Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),uminus_uminus(A,Y)) ) ) ).

% compl_le_swap1
tff(fact_2214_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,Y)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,Xa)),Y) ) ) ).

% compl_le_swap2
tff(fact_2215_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,B2)),uminus_uminus(A,A2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_2216_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,B2)),A2) ) ) ).

% minus_less_iff
tff(fact_2217_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),uminus_uminus(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),uminus_uminus(A,A2)) ) ) ).

% less_minus_iff
tff(fact_2218_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),uminus_uminus(A,Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),uminus_uminus(A,Y)) ) ) ).

% compl_less_swap1
tff(fact_2219_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,Y)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,Xa)),Y) ) ) ).

% compl_less_swap2
tff(fact_2220_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: 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),uminus_uminus(A,B2)),uminus_uminus(A,A2)) ) ).

% add.inverse_distrib_swap
tff(fact_2221_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) )
         => ( uminus_uminus(A,A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,K)),uminus_uminus(A,A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_2222_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,A2)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),uminus_uminus(A,B2)) ) ).

% minus_mult_commute
tff(fact_2223_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 = uminus_uminus(A,B2) ) ) ) ) ).

% square_eq_iff
tff(fact_2224_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,minus_minus(A,uminus_uminus(A,B2)),A2) = aa(A,A,minus_minus(A,uminus_uminus(A,A2)),B2) ) ).

% minus_diff_commute
tff(fact_2225_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),Xa) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( Xa = Y )
            | ( Xa = uminus_uminus(A,Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2226_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ej(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),uminus_uminus(set(A),aa(fun(A,$o),set(A),collect(A),P))),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_imp_eq
tff(fact_2227_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),uminus_uminus(set(A),Ta)) = uminus_uminus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_Compl
tff(fact_2228_set__take__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_take_subset
tff(fact_2229_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Nb: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,Nb,Xs)) ) ) ).

% sorted_take
tff(fact_2230_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set(A)] : ~ aa(set(option(A)),$o,member(option(A),none(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),A3)) ).

% None_notin_image_Some
tff(fact_2231_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 = uminus_uminus(A,A2) ) ) ) ).

% add_eq_0_iff
tff(fact_2232_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),uminus_uminus(A,A2)),A2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_2233_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) )
         => ( uminus_uminus(A,A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_2234_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = 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_2235_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: 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_2236_zero__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( zero_zero(A) != uminus_uminus(A,one_one(A)) ) ) ).

% zero_neq_neg_one
tff(fact_2237_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),uminus_uminus(A,one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_2238_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_2239_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_2240_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),uminus_uminus(A,one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_2241_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xa: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa) = one_one(A) )
        <=> ( ( Xa = one_one(A) )
            | ( Xa = uminus_uminus(A,one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_2242_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),uminus_uminus(A,K)),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2243_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),uminus_uminus(A,B2)) ) ).

% diff_conv_add_uminus
tff(fact_2244_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),uminus_uminus(A,B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2245_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),Xa: A,Y: B,Xs: list(A),Ys: list(B)] :
      map_upds(A,B,fun_upd(A,option(B),F,Xa,aa(B,option(B),some(B),Y)),Xs,Ys) = $ite(aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),map_upds(A,B,F,Xs,Ys),fun_upd(A,option(B),map_upds(A,B,F,Xs,Ys),Xa,aa(B,option(B),some(B),Y))) ).

% map_upd_upds_conv_if
tff(fact_2246_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,A2)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_minus_self
tff(fact_2247_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_2248_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,A2)),B2) ) ) ).

% abs_le_D2
tff(fact_2249_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,A2)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2) ) ) ) ).

% abs_leI
tff(fact_2250_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,A2)),B2) ) ) ) ).

% abs_less_iff
tff(fact_2251_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),uminus_uminus(set(A),A3))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_2252_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) = uminus_uminus(A,sgn_sgn(A,B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_2253_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,uminus_uminus(A,one_one(A))) = uminus_uminus(A,one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_2254_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,M,Xs))),aa(list(A),set(A),set2(A),take(A,Nb,Xs))) ) ).

% set_take_subset_set_take
tff(fact_2255_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_2256_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),uminus_uminus(A,one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_2257_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_2258_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),uminus_uminus(A,one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_2259_in__image__insert__iff,axiom,
    ! [A: $tType,B4: set(set(A)),Xa: A,A3: set(A)] :
      ( ! [C8: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),C8),B4)
         => ~ aa(set(A),$o,member(A,Xa),C8) )
     => ( aa(set(set(A)),$o,member(set(A),A3),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,Xa)),B4))
      <=> ( aa(set(A),$o,member(A,Xa),A3)
          & aa(set(set(A)),$o,member(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_2260_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(A,A,abs_abs(A),A2))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2261_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & ( ( B2 = A2 )
              | ( B2 = uminus_uminus(A,A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2262_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A2 = B2 )
              | ( A2 = uminus_uminus(A,B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2263_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X4: A] :
          aa(A,A,abs_abs(A),X4) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),zero_zero(A)),uminus_uminus(A,X4),X4) ) ).

% abs_if_raw
tff(fact_2264_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = uminus_uminus(A,A2) ) ) ) ).

% abs_of_neg
tff(fact_2265_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)),uminus_uminus(A,A2),A2) ) ).

% abs_if
tff(fact_2266_Compl__insert,axiom,
    ! [A: $tType,Xa: A,A3: set(A)] : uminus_uminus(set(A),aa(set(A),set(A),insert(A,Xa),A3)) = aa(set(A),set(A),minus_minus(set(A),uminus_uminus(set(A),A3)),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_2267_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
         => ( take(A,K,Xs) = take(A,K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_2268_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = uminus_uminus(A,one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_2269_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          sgn_sgn(A,Xa) = $ite(
            Xa = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa),one_one(A),uminus_uminus(A,one_one(A))) ) ) ).

% sgn_if
tff(fact_2270_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
         => ( comm_s3205402744901411588hammer(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_2271_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,uminus_uminus(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2272_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,aa(nat,fun(nat,$o),ord_less(nat),K3),Nb)
              & ( A2 = uminus_uminus(A,aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2273_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( comm_s3205402744901411588hammer(A,uminus_uminus(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2274_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Nb: nat,Y: B] : zip(A,B,Xs,replicate(B,Nb,Y)) = map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ee(B,fun(A,product_prod(A,B))),Y),take(A,Nb,Xs)) ).

% zip_replicate2
tff(fact_2275_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_ek(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_2276_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),Xa: A] : restrict_map(A,B,F,uminus_uminus(set(A),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))) = fun_upd(A,option(B),F,Xa,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_2277_map__upd__eq__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A] : fun_upd(A,option(B),M,Xa,none(B)) = restrict_map(A,B,M,uminus_uminus(set(A),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))) ).

% map_upd_eq_restrict
tff(fact_2278_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,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),uminus_uminus(A,R2)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2279_max__ins__scaled,axiom,
    ! [Nb: nat,X14: vEBT_VEBT,M: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,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),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(nat),set(nat),insert(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))))) ).

% max_ins_scaled
tff(fact_2280_height__compose__list,axiom,
    ! [Ta: vEBT_VEBT,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,Ta),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ) ).

% height_compose_list
tff(fact_2281_dbl__dec__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,zero_zero(A)) = uminus_uminus(A,one_one(A)) ) ) ).

% dbl_dec_simps(2)
tff(fact_2282_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_2283_last__take__nth__conv,axiom,
    ! [A: $tType,Nb: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),L))
     => ( ( Nb != zero_zero(nat) )
       => ( last(A,take(A,Nb,L)) = aa(nat,A,nth(A,L),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ) ).

% last_take_nth_conv
tff(fact_2284_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = zero_zero(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Node(Uu2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_2285_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B4: set(A),Xa: B,A3: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image(A,set(product_prod(B,A)),aTP_Lamp_el(B,fun(A,set(product_prod(B,A))),Xa)),B4))),A3) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_em(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xa),A3,B4) ) ) ).

% fold_union_pair
tff(fact_2286_negative__eq__positive,axiom,
    ! [Nb: nat,M: nat] :
      ( ( 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_2287_real__add__minus__iff,axiom,
    ! [Xa: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),uminus_uminus(real,A2)) = zero_zero(real) )
    <=> ( Xa = A2 ) ) ).

% real_add_minus_iff
tff(fact_2288_negative__zle,axiom,
    ! [Nb: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),Nb))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_2289_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),uminus_uminus(word(A),one_one(word(A)))),W)
        <=> ( W = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% max_word_less_eq_iff
tff(fact_2290_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),uminus_uminus(word(A),one_one(word(A)))),Xa)
        <=> ( Xa = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% word_minus_one_le
tff(fact_2291_max__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),uminus_uminus(word(A),one_one(word(A)))) ) ).

% max_word_max
tff(fact_2292_word__n1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),uminus_uminus(word(A),one_one(word(A)))) ) ).

% word_n1_ge
tff(fact_2293_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),uminus_uminus(word(A),one_one(word(A)))),A2)
        <=> ( A2 = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% word_order.extremum_unique
tff(fact_2294_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),uminus_uminus(word(A),one_one(word(A)))) ) ).

% word_order.extremum
tff(fact_2295_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = Xa ) ).

% cSup_singleton
tff(fact_2296_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,Xa)) = Xa ) ) ) ).

% cSup_atLeastLessThan
tff(fact_2297_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xa,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_2298_negative__zless,axiom,
    ! [Nb: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(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_2299_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = Xa ) ).

% Max_singleton
tff(fact_2300_last__replicate,axiom,
    ! [A: $tType,Nb: nat,Xa: A] :
      ( ( Nb != zero_zero(nat) )
     => ( last(A,replicate(A,Nb,Xa)) = Xa ) ) ).

% last_replicate
tff(fact_2301_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),C2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_en(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cSUP_const
tff(fact_2302_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [A4: A] :
            ( aa(set(A),$o,member(A,A4),A3)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B4,A4)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% finite_UN_I
tff(fact_2303_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xa)
            <=> ! [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_2304_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xa)
            <=> ! [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa) ) ) ) ) ) ).

% Max_less_iff
tff(fact_2305_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A3: set(A),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,aTP_Lamp_eo(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_2306_set__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set2(A),concat(A,Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_2307_Some__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A3)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),A3)) ) ) ) ).

% Some_Sup
tff(fact_2308_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).

% Max_Sup
tff(fact_2309_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = aa(set(A),A,lattic643756798349783984er_Max(A),X5) ) ) ) ) ).

% cSup_eq_Max
tff(fact_2310_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Z),X5)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_2311_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) )
         => ( ! [Y3: A] :
                ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_2312_uminus__int__code_I1_J,axiom,
    uminus_uminus(int,zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_2313_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : uminus_uminus(set(A),A3) = aa(fun(A,$o),set(A),collect(A),uminus_uminus(fun(A,$o),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3))) ).

% uminus_set_def
tff(fact_2314_Some__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(B,option(B),some(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image(A,option(B),aTP_Lamp_ep(fun(A,B),fun(A,option(B)),F)),A3)) ) ) ) ).

% Some_SUP
tff(fact_2315_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) )
           => ( ! [Y3: A] :
                  ( ! [X4: A] :
                      ( aa(set(A),$o,member(A,X4),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_2316_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z) ) ) ) ).

% cSup_least
tff(fact_2317_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,Xa),X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ).

% le_cSup_finite
tff(fact_2318_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
         => ( ( X5 != bot_bot(set(A)) )
           => ~ ! [X: A] :
                  ( aa(set(A),$o,member(A,X),X5)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ) ).

% less_cSupE
tff(fact_2319_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X5))
           => ? [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X) ) ) ) ) ).

% less_cSupD
tff(fact_2320_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Xa: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,Xa),X5)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A2) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A2) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_2321_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max.coboundedI
tff(fact_2322_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => ? [Xa2: A] :
                      ( aa(set(A),$o,member(A,Xa2),B4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2) ) )
             => ( ! [X: A] :
                    ( aa(set(A),$o,member(A,X),B4)
                   => ? [Xa2: A] :
                        ( aa(set(A),$o,member(A,Xa2),A3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,lattic643756798349783984er_Max(A),B4) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_2323_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [Y3: A] :
                ( aa(set(A),$o,member(A,Y3),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
           => ( aa(set(A),$o,member(A,Xa),A3)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = Xa ) ) ) ) ) ).

% Max_eqI
tff(fact_2324_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,Xa),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max_ge
tff(fact_2325_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3) ) ) ) ).

% Max_in
tff(fact_2326_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,$o,P,uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))
       => aa(int,$o,P,Z) ) ) ).

% int_of_nat_induct
tff(fact_2327_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z != uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% int_cases
tff(fact_2328_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_2329_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,minus_minus(int,zero_zero(int)),L) = uminus_uminus(int,L) ).

% minus_int_code(2)
tff(fact_2330_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),uminus_uminus(word(A),one_one(word(A)))),A2)
         => ( A2 = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% word_order.extremum_uniqueI
tff(fact_2331_max__word__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( uminus_uminus(word(A),one_one(word(A))) != zero_zero(word(A)) ) ) ).

% max_word_not_0
tff(fact_2332_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( A2 != uminus_uminus(word(A),one_one(word(A))) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),uminus_uminus(word(A),one_one(word(A)))) ) ) ).

% word_order.not_eq_extremum
tff(fact_2333_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),uminus_uminus(word(A),one_one(word(A)))),A2) ) ).

% word_order.extremum_strict
tff(fact_2334_word__not__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),uminus_uminus(word(A),one_one(word(A)))),Y) ) ).

% word_not_simps(3)
tff(fact_2335_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F: fun(A,B),M6: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),M6) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))),M6) ) ) ) ).

% cSUP_least
tff(fact_2336_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A2)
            <=> ! [X3: A] :
                  ( aa(set(A),$o,member(A,X3),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_2337_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S3))),A2) ) ) ) ).

% cSup_abs_le
tff(fact_2338_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => ( aa(set(A),$o,member(A,Y3),A3)
                   => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y3)),A3) ) )
             => aa(set(A),$o,member(A,aa(set(A),A,complete_Sup_Sup(A),A3)),A3) ) ) ) ) ).

% finite_Sup_in
tff(fact_2339_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xa) ) ) ) ) ).

% Max.boundedI
tff(fact_2340_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xa)
             => ! [A11: A] :
                  ( aa(set(A),$o,member(A,A11),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A11),Xa) ) ) ) ) ) ).

% Max.boundedE
tff(fact_2341_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A3) )
            <=> ( aa(set(A),$o,member(A,M),A3)
                & ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2342_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X3) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_2343_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = M )
            <=> ( aa(set(A),$o,member(A,M),A3)
                & ! [X3: A] :
                    ( aa(set(A),$o,member(A,X3),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2344_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X3) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_2345_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [B3: A] :
                ( aa(set(A),$o,member(A,B3),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A2) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_2346_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( M != uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_2347_real__add__less__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),uminus_uminus(real,Xa)) ) ).

% real_add_less_0_iff
tff(fact_2348_real__0__less__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,Xa)),Y) ) ).

% real_0_less_add_iff
tff(fact_2349_real__0__le__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,Xa)),Y) ) ).

% real_0_le_add_iff
tff(fact_2350_real__add__le__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),uminus_uminus(real,Xa)) ) ).

% real_add_le_0_iff
tff(fact_2351_int__zle__neg,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),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_2352_abs__real__def,axiom,
    ! [A2: real] :
      aa(real,real,abs_abs(real),A2) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real)),uminus_uminus(real,A2),A2) ).

% abs_real_def
tff(fact_2353_negative__zle__0,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),Nb))),zero_zero(int)) ).

% negative_zle_0
tff(fact_2354_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ~ ! [N: nat] : K != uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nonpos_int_cases
tff(fact_2355_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),Xa)
        <=> ( Xa = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% plus_1_less
tff(fact_2356_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),P3)),P3)
        <=> ( P3 = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% overflow_plus_one_self
tff(fact_2357_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),uminus_uminus(int,I),I) ).

% zabs_def
tff(fact_2358_max__word__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) )
         => ( Xa = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% max_word_wrap
tff(fact_2359_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),uminus_uminus(word(A),Y))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) ) ) ).

% no_plus_overflow_neg
tff(fact_2360_finite__subset__Union,axiom,
    ! [A: $tType,A3: set(A),B9: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B9))
       => ~ ! [F8: set(set(A))] :
              ( aa(set(set(A)),$o,finite_finite2(set(A)),F8)
             => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F8),B9)
               => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F8)) ) ) ) ) ).

% finite_subset_Union
tff(fact_2361_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(set(A),A,complete_Sup_Sup(A),S3)),L))),E2) ) ) ) ).

% cSup_asclose
tff(fact_2362_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,complete_Sup_Sup(A),A3) = finite_fold(A,A,sup_sup(A),bot_bot(A),A3) ) ) ) ).

% Sup_fold_sup
tff(fact_2363_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B4)) ) ) ) ) ).

% Max.subset_imp
tff(fact_2364_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M6),N7)
         => ( ( M6 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N7)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M6)),aa(set(A),A,lattic643756798349783984er_Max(A),N7)) ) ) ) ) ).

% Max_mono
tff(fact_2365_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) )
       => ~ ! [N: nat] :
              ( ( K = uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),N)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_2366_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)))) ).

% not_zle_0_negative
tff(fact_2367_negative__zless__0,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(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_2368_negD,axiom,
    ! [Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),zero_zero(int))
     => ? [N: nat] : Xa = uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_2369_word__le__make__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != uminus_uminus(word(A),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),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_2370_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Xa: word(A)] :
          ( ( K != uminus_uminus(word(A),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),K) ) ) ) ).

% word_Suc_leq
tff(fact_2371_word__Suc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),K: word(A)] :
          ( ( Xa != uminus_uminus(word(A),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),K)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),K) ) ) ) ).

% word_Suc_le
tff(fact_2372_foldl__set,axiom,
    ! [A: $tType,L: list(set(A))] : foldl(set(A),set(A),sup_sup(set(A)),bot_bot(set(A)),L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_eq(list(set(A)),fun(set(A),$o),L))) ).

% foldl_set
tff(fact_2373_zsgn__def,axiom,
    ! [I: int] :
      sgn_sgn(int,I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),uminus_uminus(int,one_one(int))) ) ).

% zsgn_def
tff(fact_2374_sgn__real__def,axiom,
    ! [A2: real] :
      sgn_sgn(real,A2) = $ite(
        A2 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2),one_one(real),uminus_uminus(real,one_one(real))) ) ).

% sgn_real_def
tff(fact_2375_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_er(fun(A,B),fun(B,fun(A,B)),F),K)),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,F),S3))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_2376_Id__on__def,axiom,
    ! [A: $tType,A3: set(A)] : id_on(A,A3) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image(A,set(product_prod(A,A)),aTP_Lamp_es(A,set(product_prod(A,A)))),A3)) ).

% Id_on_def
tff(fact_2377_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K = uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),N)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_2378_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(uminus_uminus(int,A2),B2,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R2 = zero_zero(int),uminus_uminus(int,Q4),aa(int,int,minus_minus(int,uminus_uminus(int,Q4)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_2379_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)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_2380_UN__insert,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A2: B,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(B),set(B),insert(B,A2),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B4,A2)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) ).

% UN_insert
tff(fact_2381_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B4: set(A),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_et(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4)) ).

% UN_simps(2)
tff(fact_2382_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_eu(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3)))) ).

% UN_simps(3)
tff(fact_2383_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B4: fun(B,set(A)),C3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ev(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B4)),C3)) = $ite(C3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert(A,A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3)))) ).

% UN_simps(1)
tff(fact_2384_UN__singleton,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image(A,set(A),aTP_Lamp_ew(A,set(A))),A3)) = A3 ).

% UN_singleton
tff(fact_2385_ccSUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),bot_bot(set(B)))) = bot_bot(A) ) ).

% ccSUP_empty
tff(fact_2386_subset__mset_OcSup__singleton,axiom,
    ! [A: $tType,Xa: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),insert(multiset(A),Xa),bot_bot(set(multiset(A))))) = Xa ).

% subset_mset.cSup_singleton
tff(fact_2387_subset__mset_OcSUP__const,axiom,
    ! [A: $tType,B: $tType,A3: set(A),C2: multiset(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image(A,multiset(B),aTP_Lamp_ex(multiset(B),fun(A,multiset(B)),C2)),A3)) = C2 ) ) ).

% subset_mset.cSUP_const
tff(fact_2388_Sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [A3: set(fun(B,A)),Xa: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),A3),Xa) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,A)),set(A),image(fun(B,A),A,aTP_Lamp_ey(B,fun(fun(B,A),A),Xa)),A3)) ) ).

% Sup_apply
tff(fact_2389_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = bot_bot(A) )
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),A3)
             => ( X3 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_2390_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A3) )
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),A3)
             => ( X3 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_2391_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_2392_SUP__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image(A,A,aTP_Lamp_ez(A,A)),A3)) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ).

% SUP_identity_eq
tff(fact_2393_SUP__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [F: fun(C,fun(B,A)),A3: set(C),Xa: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),aa(set(C),set(fun(B,A)),image(C,fun(B,A),F),A3)),Xa) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_fa(fun(C,fun(B,A)),fun(B,fun(C,A)),F),Xa)),A3)) ) ).

% SUP_apply
tff(fact_2394_UN__I,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B2: B,B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(B),$o,member(B,B2),aa(A,set(B),B4,A2))
       => aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% UN_I
tff(fact_2395_UN__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
    <=> ? [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
          & aa(set(A),$o,member(A,B2),aa(B,set(A),B4,X3)) ) ) ).

% UN_iff
tff(fact_2396_empty__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),bot_bot(set(option(A)))) = none(A) ) ) ).

% empty_Sup
tff(fact_2397_singleton__None__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A))))) = none(A) ) ) ).

% singleton_None_Sup
tff(fact_2398_Sup__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).

% Sup_empty
tff(fact_2399_ccSup__empty,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).

% ccSup_empty
tff(fact_2400_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B4),A3)) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
             => ( aa(B,A,B4,X3) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_2401_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B4),A3)) = bot_bot(A) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
             => ( aa(B,A,B4,X3) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_2402_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_fb(B,A)),A3)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_2403_ccSUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_fc(B,A)),A3)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_2404_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_fd(B,fun(A,B),F)),A3)) = F ) ) ) ).

% SUP_const
tff(fact_2405_ccSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_fe(B,fun(A,B),F)),A3)) = F ) ) ) ).

% ccSUP_const
tff(fact_2406_UN__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_ff(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),C2) ).

% UN_constant
tff(fact_2407_UN__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A3: set(B),B4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),A3))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),B4))) ).

% UN_Un
tff(fact_2408_Sup__set__def,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fg(set(set(A)),fun(A,$o),A3)) ).

% Sup_set_def
tff(fact_2409_subset__mset_OcSup__eq__maximum,axiom,
    ! [A: $tType,Z: multiset(A),X5: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,member(multiset(A),Z),X5)
     => ( ! [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),X5)
           => subseteq_mset(A,X,Z) )
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = Z ) ) ) ).

% subset_mset.cSup_eq_maximum
tff(fact_2410_SUP__Sup__eq,axiom,
    ! [A: $tType,S3: set(set(A)),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o))),S3)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S3)) ) ).

% SUP_Sup_eq
tff(fact_2411_subset__mset_OcSUP__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(A,multiset(B)),M6: multiset(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => subseteq_mset(B,aa(A,multiset(B),F,X),M6) )
       => subseteq_mset(B,aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image(A,multiset(B),F),A3)),M6) ) ) ).

% subset_mset.cSUP_least
tff(fact_2412_subset__mset_OcSup__least,axiom,
    ! [A: $tType,X5: set(multiset(A)),Z: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),X5)
           => subseteq_mset(A,X,Z) )
       => subseteq_mset(A,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5),Z) ) ) ).

% subset_mset.cSup_least
tff(fact_2413_subset__mset_OcSup__eq__non__empty,axiom,
    ! [A: $tType,X5: set(multiset(A)),A2: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),X5)
           => subseteq_mset(A,X,A2) )
       => ( ! [Y3: multiset(A)] :
              ( ! [X4: multiset(A)] :
                  ( aa(set(multiset(A)),$o,member(multiset(A),X4),X5)
                 => subseteq_mset(A,X4,Y3) )
             => subseteq_mset(A,A2,Y3) )
         => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = A2 ) ) ) ) ).

% subset_mset.cSup_eq_non_empty
tff(fact_2414_Sup__multiset__empty,axiom,
    ! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).

% Sup_multiset_empty
tff(fact_2415_subset__mset_Ole__cSup__finite,axiom,
    ! [A: $tType,X5: set(multiset(A)),Xa: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),X5)
     => ( aa(set(multiset(A)),$o,member(multiset(A),Xa),X5)
       => subseteq_mset(A,Xa,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5)) ) ) ).

% subset_mset.le_cSup_finite
tff(fact_2416_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S3: set(B),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),aTP_Lamp_fh(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S3)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),R2),S3))) ) ).

% SUP_UN_eq
tff(fact_2417_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o)))),S3)),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S3)) ) ).

% SUP_Sup_eq2
tff(fact_2418_Sup__nat__def,axiom,
    ! [X5: set(nat)] :
      aa(set(nat),nat,complete_Sup_Sup(nat),X5) = $ite(X5 = bot_bot(set(nat)),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),X5)) ).

% Sup_nat_def
tff(fact_2419_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S3: set(C),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_fi(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S3)),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R2),S3))) ) ).

% SUP_UN_eq2
tff(fact_2420_Inf_OINF__identity__eq,axiom,
    ! [A: $tType,Inf: fun(set(A),A),A3: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),image(A,A,aTP_Lamp_bp(A,A)),A3)) = aa(set(A),A,Inf,A3) ).

% Inf.INF_identity_eq
tff(fact_2421_Sup_OSUP__identity__eq,axiom,
    ! [A: $tType,Sup: fun(set(A),A),A3: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),image(A,A,aTP_Lamp_bp(A,A)),A3)) = aa(set(A),A,Sup,A3) ).

% Sup.SUP_identity_eq
tff(fact_2422_Sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A3: set(fun(A,B)),X4: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A3),X4) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image(fun(A,B),B,aTP_Lamp_fj(A,fun(fun(A,B),B),X4)),A3)) ) ).

% Sup_fun_def
tff(fact_2423_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),Va2: A] :
          ( aa(set(A),$o,member(A,U),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Va2),U)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Va2),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% Sup_upper2
tff(fact_2424_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2)
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B2) ) ) ) ).

% Sup_le_iff
tff(fact_2425_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,Xa),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Sup_upper
tff(fact_2426_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z) ) ) ).

% Sup_least
tff(fact_2427_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( ! [A4: A] :
              ( aa(set(A),$o,member(A,A4),A3)
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),B4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X4) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ).

% Sup_mono
tff(fact_2428_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Xa: A] :
          ( ! [Y3: A] :
              ( aa(set(A),$o,member(A,Y3),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
         => ( ! [Y3: A] :
                ( ! [Z4: A] :
                    ( aa(set(A),$o,member(A,Z4),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z4),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A3) = Xa ) ) ) ) ).

% Sup_eqI
tff(fact_2429_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S3))
        <=> ? [X3: A] :
              ( aa(set(A),$o,member(A,X3),S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X3) ) ) ) ).

% less_Sup_iff
tff(fact_2430_empty__Union__conv,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) )
    <=> ! [X3: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X3),A3)
         => ( X3 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_2431_Union__empty__conv,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = bot_bot(set(A)) )
    <=> ! [X3: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X3),A3)
         => ( X3 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_2432_Union__empty,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),bot_bot(set(set(A)))) = bot_bot(set(A)) ).

% Union_empty
tff(fact_2433_Union__subsetI,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(set(A))] :
      ( ! [X: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X),A3)
         => ? [Y4: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),Y4),B4)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Y4) ) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) ) ).

% Union_subsetI
tff(fact_2434_Union__upper,axiom,
    ! [A: $tType,B4: set(A),A3: set(set(A))] :
      ( aa(set(set(A)),$o,member(set(A),B4),A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) ) ).

% Union_upper
tff(fact_2435_Union__least,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X7),A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),C3) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),C3) ) ).

% Union_least
tff(fact_2436_Union__mono,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A3),B4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) ) ).

% Union_mono
tff(fact_2437_SUP__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aTP_Lamp_fk(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),B4)),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(set(B),fun(C,A),aTP_Lamp_fm(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A3)),B4)) ) ).

% SUP_commute
tff(fact_2438_UN__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: fun(C,set(A)),B4: fun(B,set(C)),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_fn(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C3),B4)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),C3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image(B,set(C),B4),A3)))) ).

% UN_extend_simps(9)
tff(fact_2439_UN__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
     => ~ ! [X: B] :
            ( aa(set(B),$o,member(B,X),A3)
           => ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B4,X)) ) ) ).

% UN_E
tff(fact_2440_UN__UN__flatten,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(B,set(A)),B4: fun(C,set(B)),A3: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),C3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),B4),A3)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_fo(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C3),B4)),A3)) ).

% UN_UN_flatten
tff(fact_2441_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A3))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
             => ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X3) ) ) ) ) ).

% le_Sup_iff
tff(fact_2442_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B4: set(B),F: fun(A,C),G: fun(B,C)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F,I2)),aa(B,C,G,X4)) ) )
         => ( ! [J2: B] :
                ( aa(set(B),$o,member(B,J2),B4)
               => ? [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,J2)),aa(A,C,F,X4)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F),A3)) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G),B4)) ) ) ) ) ).

% SUP_eq
tff(fact_2443_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V2: A] :
              ( aa(set(A),$o,member(A,V2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2) )
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% less_eq_Sup
tff(fact_2444_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ).

% Sup_subset_mono
tff(fact_2445_SUP__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => ( aa(A,B,F,I2) = Xa ) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),I5)) = Xa ) ) ) ) ).

% SUP_eq_const
tff(fact_2446_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F: fun(A,B),Xa: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),Xa) )
         => ( ! [Y3: B] :
                ( ! [I4: A] :
                    ( aa(set(A),$o,member(A,I4),A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I4)),Y3) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),Y3) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3)) = Xa ) ) ) ) ).

% SUP_eqI
tff(fact_2447_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B4: set(B),F: fun(A,C),G: fun(B,C)] :
          ( ! [N: A] :
              ( aa(set(A),$o,member(A,N),A3)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F,N)),aa(B,C,G,X4)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F),A3))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G),B4))) ) ) ).

% SUP_mono
tff(fact_2448_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))),U) ) ) ).

% SUP_least
tff(fact_2449_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% SUP_mono'
tff(fact_2450_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))) ) ) ).

% SUP_upper
tff(fact_2451_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3))),U)
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X3)),U) ) ) ) ).

% SUP_le_iff
tff(fact_2452_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),U: B,F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F,I))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))) ) ) ) ).

% SUP_upper2
tff(fact_2453_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: set(B),Y: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3))),Y)
         => ( aa(set(B),$o,member(B,I),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_2454_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3)))
        <=> ? [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F,X3)) ) ) ) ).

% less_SUP_iff
tff(fact_2455_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),A3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fp(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A3)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_2456_SUP__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I5: set(A),A3: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I5)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,A3,K)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,A3),I5))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,A3),I5)) ) ) ) ).

% SUP_absorb
tff(fact_2457_UN__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,set(A)),F: fun(B,C),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_fq(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B4),F)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),aa(set(B),set(C),image(B,C,F),A3))) ).

% UN_extend_simps(10)
tff(fact_2458_image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),B4: fun(C,set(B)),A3: set(C)] : aa(set(B),set(A),image(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),B4),A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_fr(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F),B4)),A3)) ).

% image_UN
tff(fact_2459_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_fs(B,set(A))),A3)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_2460_UN__empty,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_2461_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) )
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
         => ( aa(B,set(A),B4,X3) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_2462_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) = bot_bot(set(A)) )
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
         => ( aa(B,set(A),B4,X3) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_2463_UN__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F,X)),aa(A,set(B),G,X)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F),A3))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),B4))) ) ) ).

% UN_mono
tff(fact_2464_UN__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(A,set(B)),C3: set(B)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B4,X)),C3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))),C3) ) ).

% UN_least
tff(fact_2465_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B4,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ).

% UN_upper
tff(fact_2466_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I5: set(B),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),B4)
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),A3,X3)),B4) ) ) ).

% UN_subset_iff
tff(fact_2467_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A3: set(A),A2: B,B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A3)
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ft(B,fun(fun(A,set(B)),fun(A,set(B))),A2),B4)),A3)) = aa(set(B),set(B),insert(B,A2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% UN_insert_distrib
tff(fact_2468_UN__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B4: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_fu(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3)) ).

% UN_extend_simps(6)
tff(fact_2469_UN__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A3,K)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)) ) ) ).

% UN_absorb
tff(fact_2470_UN__Un__distrib,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fv(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),I5))) ).

% UN_Un_distrib
tff(fact_2471_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),C3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fv(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) ).

% Un_Union_image
tff(fact_2472_image__Union,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),S3: set(set(B))] : aa(set(B),set(A),image(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F)),S3)) ).

% image_Union
tff(fact_2473_UN__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(set(B))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image(set(B),set(A),aTP_Lamp_fw(fun(B,set(A)),fun(set(B),set(A)),B4)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A3))) ).

% UN_extend_simps(8)
tff(fact_2474_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,F: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3)))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),aa(B,A,F,X3)) ) ) ) ) ).

% le_SUP_iff
tff(fact_2475_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),C2: B,F: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F,I2)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),I5)) = C2 )
            <=> ! [X3: A] :
                  ( aa(set(A),$o,member(A,X3),I5)
                 => ( aa(A,B,F,X3) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2476_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),B4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),B4))) ) ) ) ).

% SUP_subset_mono
tff(fact_2477_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A3: set(B)] :
          aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_fx(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(A),C2) ) ).

% SUP_constant
tff(fact_2478_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_2479_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A2: B,A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),insert(B,A2),A3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,A2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3))) ) ).

% SUP_insert
tff(fact_2480_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A3: set(B),B4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M6),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M6),B4))) ) ).

% SUP_union
tff(fact_2481_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B4: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),insert(A,A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = $ite(C3 = bot_bot(set(B)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ev(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B4)),C3))) ).

% UN_extend_simps(1)
tff(fact_2482_SUP__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),G: fun(C,set(B)),A3: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),G),A3)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_fy(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F),G)),A3)) ) ).

% SUP_UNION
tff(fact_2483_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = $ite(C3 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_eu(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3))) ).

% UN_extend_simps(3)
tff(fact_2484_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4) = $ite(C3 = bot_bot(set(B)),B4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_et(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3))) ).

% UN_extend_simps(2)
tff(fact_2485_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_fz(fun(B,A),fun(B,set(A)),F)),A3)) = aa(set(B),set(A),image(B,A,F),A3) ).

% UNION_singleton_eq_range
tff(fact_2486_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I: B,B4: set(A),J4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),fun_upd(B,set(A),A3,I,B4)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),aa(set(B),set(B),minus_minus(set(B),J4),aa(set(B),set(B),insert(B,I),bot_bot(set(B))))))),
        $ite(aa(set(B),$o,member(B,I),J4),B4,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_2487_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xa: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = Xa ) ).

% ccpo_Sup_singleton
tff(fact_2488_max__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Nb: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,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,aa(nat,fun(nat,nat),ord_max(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13)))))))) ) ).

% max_idx_list
tff(fact_2489_height__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Foo),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))) ) ).

% height_i_max
tff(fact_2490_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_ga(A,fun(nat,A),B4)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B4) ) ).

% SUP_nat_binary
tff(fact_2491_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F),bot_bot(set(B))))) = A3 ).

% Union_image_empty
tff(fact_2492_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),G: fun(C,set(B)),Xa: C,X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F),aa(C,set(B),G,Xa)))),X5)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),G,Xa)),aa(fun(B,$o),set(B),collect(B),aa(set(A),fun(B,$o),aTP_Lamp_gb(fun(B,set(A)),fun(set(A),fun(B,$o)),F),X5))) ) ).

% UN_image_subset
tff(fact_2493_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_2494_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_2495_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.bounded_iff
tff(fact_2496_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_2497_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_2498_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb3
tff(fact_2499_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),bot_bot(A)) = Xa ) ).

% max_bot2
tff(fact_2500_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),Xa) = Xa ) ).

% max_bot
tff(fact_2501_max__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Nb)) ).

% max_Suc_Suc
tff(fact_2502_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(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_2503_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_2504_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_2505_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_2506_max__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Nb) = Nb ).

% max_0L
tff(fact_2507_max__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),zero_zero(nat)) = Nb ).

% max_0R
tff(fact_2508_max__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),one_one(A)) = one_one(A) ) ) ).

% max_0_1(1)
tff(fact_2509_max__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),zero_zero(A)) = one_one(A) ) ) ).

% max_0_1(2)
tff(fact_2510_SUP2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A2: A,A3: set(A),B4: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B4,A2),B2),C2)
       => aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(A),set(fun(B,fun(C,$o))),image(A,fun(B,fun(C,$o)),B4),A3)),B2),C2) ) ) ).

% SUP2_I
tff(fact_2511_SUP1__I,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),B4: fun(A,fun(B,$o)),B2: B] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(B,$o,aa(A,fun(B,$o),B4,A2),B2)
       => aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Sup_Sup(fun(B,$o)),aa(set(A),set(fun(B,$o)),image(A,fun(B,$o),B4),A3)),B2) ) ) ).

% SUP1_I
tff(fact_2512_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,Xa),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ).

% Max_insert
tff(fact_2513_SUP2__E,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B4),A3)),B2),C2)
     => ~ ! [X: C] :
            ( aa(set(C),$o,member(C,X),A3)
           => ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B4,X),B2),C2) ) ) ).

% SUP2_E
tff(fact_2514_SUP1__E,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,fun(A,$o)),A3: set(B),B2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B4),A3)),B2)
     => ~ ! [X: B] :
            ( aa(set(B),$o,member(B,X),A3)
           => ~ aa(A,$o,aa(B,fun(A,$o),B4,X),B2) ) ) ).

% SUP1_E
tff(fact_2515_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_2516_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_2517_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_2518_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_2519_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_2520_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_2521_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_2522_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded2
tff(fact_2523_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded1
tff(fact_2524_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.order_iff
tff(fact_2525_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2) ) ) ) ).

% max.boundedI
tff(fact_2526_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.boundedE
tff(fact_2527_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% max.orderI
tff(fact_2528_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_2529_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_2530_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_2531_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Xa ) ) ) ).

% max_absorb1
tff(fact_2532_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),B2,A2) ) ).

% max_def
tff(fact_2533_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_2534_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_2535_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_2536_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_2537_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_2538_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_2539_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_2540_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,minus_minus(A,Xa),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_2541_nat__add__max__left,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Nb)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Q4)) ).

% nat_add_max_left
tff(fact_2542_nat__add__max__right,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(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),Q4)) ).

% nat_add_max_right
tff(fact_2543_nat__mult__max__left,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Nb)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ).

% nat_mult_max_left
tff(fact_2544_nat__mult__max__right,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(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),Q4)) ).

% nat_mult_max_right
tff(fact_2545_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa2),Xa2,X4) ) ).

% max_def_raw
tff(fact_2546_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,aa(nat,fun(nat,nat),ord_max(nat),Nb),M) ).

% nat_minus_add_max
tff(fact_2547_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ar(nat,fun(nat,$o)),aTP_Lamp_as(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_2548_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert(A,Xa),S3)) = $ite(S3 = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,complete_Sup_Sup(A),S3))) ) ) ) ).

% Sup_insert_finite
tff(fact_2549_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X)),aa(A,A,H,Y3))
         => ( aa(set(A),$o,finite_finite2(A),N7)
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798349783984er_Max(A),N7)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image(A,A,H),N7)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_2550_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( B4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B4)),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ) ).

% Max.subset
tff(fact_2551_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] : aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)),aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y3),bot_bot(set(A)))))
             => aa(set(A),$o,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3) ) ) ) ) ).

% Max.closed
tff(fact_2552_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,Xa),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,Xa),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_2553_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B4)
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B4)) ) ) ) ) ) ) ).

% Max.union
tff(fact_2554_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_au(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% conj_subset_def
tff(fact_2555_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,Xa),A3)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_2556_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,Xa),A3)) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_2557_Sup__option__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(option(A))] :
          aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A3) = $ite(
            ( ( A3 = bot_bot(set(option(A))) )
            | ( A3 = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ),
            none(A),
            aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),these(A,A3))) ) ) ).

% Sup_option_def
tff(fact_2558_UN__constant__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),F: fun(A,set(B)),C2: set(B)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => ( aa(A,set(B),F,X) = C2 ) )
       => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F),A3)) = C2 ) ) ) ).

% UN_constant_eq
tff(fact_2559_these__not__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) != bot_bot(set(A)) )
    <=> ( ( B4 != bot_bot(set(option(A))) )
        & ( B4 != aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_2560_these__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) = bot_bot(set(A)) )
    <=> ( ( B4 = bot_bot(set(option(A))) )
        | ( B4 = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_2561_these__insert__Some,axiom,
    ! [A: $tType,Xa: A,A3: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),aa(A,option(A),some(A),Xa)),A3)) = aa(set(A),set(A),insert(A,Xa),these(A,A3)) ).

% these_insert_Some
tff(fact_2562_these__image__Some__eq,axiom,
    ! [A: $tType,A3: set(A)] : these(A,aa(set(A),set(option(A)),image(A,option(A),some(A)),A3)) = A3 ).

% these_image_Some_eq
tff(fact_2563_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_2564_these__insert__None,axiom,
    ! [A: $tType,A3: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),none(A)),A3)) = these(A,A3) ).

% these_insert_None
tff(fact_2565_in__these__eq,axiom,
    ! [A: $tType,Xa: A,A3: set(option(A))] :
      ( aa(set(A),$o,member(A,Xa),these(A,A3))
    <=> aa(set(option(A)),$o,member(option(A),aa(A,option(A),some(A),Xa)),A3) ) ).

% in_these_eq
tff(fact_2566_Some__image__these__eq,axiom,
    ! [A: $tType,A3: set(option(A))] : aa(set(A),set(option(A)),image(A,option(A),some(A)),these(A,A3)) = aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_gc(set(option(A)),fun(option(A),$o),A3)) ).

% Some_image_these_eq
tff(fact_2567_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_gd(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_2568_Un__set__drop__extend,axiom,
    ! [A: $tType,J: nat,L: list(set(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(set(A)),nat,size_size(list(set(A))),L))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(nat,set(A),nth(set(A),L),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),J,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).

% Un_set_drop_extend
tff(fact_2569_subset__subseqs,axiom,
    ! [A: $tType,X5: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(list(A),set(A),set2(A),Xs))
     => aa(set(set(A)),$o,member(set(A),X5),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_2570_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A3: set(B),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F),A3)),B4)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F)),finite_Fpow(B,A3))),finite_Fpow(A,B4)) ) ).

% image_Fpow_mono
tff(fact_2571_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_ge(fun(B,list(A)),fun(B,set(A)),F)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_2572_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),uminus_uminus(A,one_one(A))),K)),comm_s3205402744901411588hammer(A,uminus_uminus(A,B2),K)) ) ).

% pochhammer_minus'
tff(fact_2573_power__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),Xa)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% power_shift
tff(fact_2574_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_2575_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_2576_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( Xa = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2577_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
        | ( Nb = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2578_drop0,axiom,
    ! [A: $tType,X4: list(A)] : drop(A,zero_zero(nat),X4) = X4 ).

% drop0
tff(fact_2579_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(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_2580_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_2581_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_2582_drop__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,Nb),aa(list(A),list(A),cons(A,Xa),Xs)) = drop(A,Nb,Xs) ).

% drop_Suc_Cons
tff(fact_2583_length__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),drop(A,Nb,Xs)) = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_drop
tff(fact_2584_drop__upd__irrelevant,axiom,
    ! [A: $tType,M: nat,Nb: nat,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => ( drop(A,Nb,list_update(A,L,M,Xa)) = drop(A,Nb,L) ) ) ).

% drop_upd_irrelevant
tff(fact_2585_drop__update__cancel,axiom,
    ! [A: $tType,Nb: nat,M: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
     => ( drop(A,M,list_update(A,Xs,Nb,Xa)) = drop(A,M,Xs) ) ) ).

% drop_update_cancel
tff(fact_2586_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_2587_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),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),uminus_uminus(A,one_one(A))),Nb)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_2588_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),uminus_uminus(A,one_one(A))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),uminus_uminus(A,one_one(A))),Nb)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_2589_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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% power_eq_0_iff
tff(fact_2590_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xa) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_2591_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_2592_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xa) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2593_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2594_last__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( last(A,drop(A,Nb,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_2595_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(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,aa(nat,fun(nat,$o),ord_less(nat),Nb),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2596_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_2597_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y) ) ) ) ).

% power_increasing_iff
tff(fact_2598_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb))
        <=> ( ( A2 != zero_zero(A) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2599_nth__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,drop(A,Nb,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),I)) ) ) ).

% nth_drop
tff(fact_2600_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_2601_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Nb))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2602_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_2603_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_2604_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_2605_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_2606_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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),Xa),Nb)) ) ) ) ).

% power_commuting_commutes
tff(fact_2607_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_2608_drop__eq__ConsD,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Xa: A,Xs4: list(A)] :
      ( ( drop(A,Nb,Xs) = aa(list(A),list(A),cons(A,Xa),Xs4) )
     => ( drop(A,aa(nat,nat,suc,Nb),Xs) = Xs4 ) ) ).

% drop_eq_ConsD
tff(fact_2609_set__drop__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_drop_subset
tff(fact_2610_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Nb: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,Nb,Xs)) ) ) ).

% sorted_drop
tff(fact_2611_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono
tff(fact_2612_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_le_power
tff(fact_2613_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(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_2614_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% one_le_power
tff(fact_2615_empty__in__Fpow,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(set(A)),$o,member(set(A),bot_bot(set(A))),finite_Fpow(A,A3)) ).

% empty_in_Fpow
tff(fact_2616_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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),Xa),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_2617_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_2618_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_2619_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_2620_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_2621_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_2622_Fpow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_2623_set__drop__subset__set__drop,axiom,
    ! [A: $tType,Nb: nat,M: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,M,Xs))),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))) ) ).

% set_drop_subset_set_drop
tff(fact_2624_drop__update__swap,axiom,
    ! [A: $tType,M: nat,Nb: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( drop(A,M,list_update(A,Xs,Nb,Xa)) = list_update(A,drop(A,M,Xs),aa(nat,nat,minus_minus(nat,Nb),M),Xa) ) ) ).

% drop_update_swap
tff(fact_2625_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_2626_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_2627_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_2628_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_2629_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(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_2630_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(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_2631_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(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_2632_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_2633_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N7: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N7)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(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),N7)) ) ) ) ).

% power_strict_increasing
tff(fact_2634_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(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,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ) ) ) ).

% power_less_imp_less_exp
tff(fact_2635_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(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),uminus_uminus(A,one_one(A))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_minus
tff(fact_2636_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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_2637_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N7: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N7)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N7)) ) ) ) ).

% power_increasing
tff(fact_2638_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb)) ) ).

% zero_le_power_abs
tff(fact_2639_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),K)) ) ).

% power_gt_expt
tff(fact_2640_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb)) ) ).

% nat_one_le_power
tff(fact_2641_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ? [R3: real] :
          ( aa(real,$o,aa(real,fun(real,$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_2642_real__arch__pow__inv,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_2643_drop__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] :
      drop(A,Nb,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(Nb = zero_zero(nat),aa(list(A),list(A),cons(A,Xa),Xs),drop(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xs)) ).

% drop_Cons'
tff(fact_2644_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(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_2645_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_2646_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(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_2647_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_2648_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N7: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N7)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N7)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_2649_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N7: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N7)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N7)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_2650_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2651_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2652_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% power_le_imp_le_exp
tff(fact_2653_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% self_le_power
tff(fact_2654_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(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_2655_Fpow__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),finite_Fpow(A,B4)) ) ).

% Fpow_mono
tff(fact_2656_realpow__pos__nth,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$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_2657_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X),Nb) = A2 )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y4)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),Nb) = A2 ) )
               => ( Y4 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2658_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$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_2659_Fpow__def,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_gf(set(A),fun(set(A),$o),A3)) ).

% Fpow_def
tff(fact_2660_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs)) = drop(A,I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_2661_in__set__drop__conv__nth,axiom,
    ! [A: $tType,Xa: A,Nb: nat,L: list(A)] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),drop(A,Nb,L)))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L))
          & ( Xa = aa(nat,A,nth(A,L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_2662_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
             => aa(A,$o,aa(A,fun(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_2663_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_2664_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(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_2665_sgn__power__injE,axiom,
    ! [A2: real,Nb: nat,Xa: 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),aa(real,real,abs_abs(real),A2)),Nb)) = Xa )
     => ( ( Xa = 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),aa(real,real,abs_abs(real),B2)),Nb)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_2666_foldl__list__update,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F: fun(B,fun(A,B)),A2: B,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( foldl(B,A,F,A2,list_update(A,Xs,Nb,Xa)) = foldl(B,A,F,aa(A,B,aa(B,fun(A,B),F,foldl(B,A,F,A2,take(A,Nb,Xs))),Xa),drop(A,aa(nat,nat,suc,Nb),Xs)) ) ) ).

% foldl_list_update
tff(fact_2667_linear__plus__1__le__power,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xa)),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),Xa),one_one(real))),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_2668_Bernoulli__inequality,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xa))),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)),Xa)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_2669_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(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),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_2670_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),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),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_2671_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(set(complex),$o,finite_finite2(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_gg(nat,fun(complex,fun(complex,$o)),Nb),C2))) ) ).

% finite_nth_roots
tff(fact_2672_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_gh(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_2673_drop__last__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ( drop(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = aa(list(A),list(A),cons(A,last(A,L)),nil(A)) ) ) ).

% drop_last_conv
tff(fact_2674_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A2: A,Xs: list(B)] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_gi(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_2675_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,I,A2) = append(A,take(A,I,Xs),aa(list(A),list(A),cons(A,A2),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_2676_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xa: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,aa(list(A),list(A),cons(A,Xa),nil(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xa ).

% concat_inth
tff(fact_2677_pos__n__replace,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),append(A,take(A,Nb,Xs),append(A,aa(list(A),list(A),cons(A,Y),nil(A)),drop(A,aa(nat,nat,suc,Nb),Xs)))) ) ) ).

% pos_n_replace
tff(fact_2678_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( append(A,Xs,Us) = append(A,Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_2679_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gj(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2680_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gk(fun(B,nat),fun(B,A),F)),A3) ) ).

% of_nat_sum
tff(fact_2681_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gl(fun(B,A),fun(B,A),F)),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gl(fun(B,A),fun(B,A),F)),A3) ) ).

% abs_sum_abs
tff(fact_2682_nth__repl,axiom,
    ! [A: $tType,M: nat,Xs: list(A),Nb: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( M != Nb )
         => ( aa(nat,A,nth(A,append(A,take(A,Nb,Xs),append(A,aa(list(A),list(A),cons(A,Xa),nil(A)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),Xs)))),M) = aa(nat,A,nth(A,Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_2683_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_2684_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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_2685_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),F4) = zero_zero(B) )
          <=> ! [X3: A] :
                ( aa(set(A),$o,member(A,X3),F4)
               => ( aa(A,B,F,X3) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_2686_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_2687_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_2688_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_2689_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),append(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_2690_take0,axiom,
    ! [A: $tType,X4: list(A)] : take(A,zero_zero(nat),X4) = nil(A) ).

% take0
tff(fact_2691_take__eq__Nil,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( take(A,Nb,Xs) = nil(A) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_2692_take__eq__Nil2,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,Nb,Xs) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_2693_replicate__empty,axiom,
    ! [A: $tType,Nb: nat,Xa: A] :
      ( ( replicate(A,Nb,Xa) = nil(A) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_2694_empty__replicate,axiom,
    ! [A: $tType,Nb: nat,Xa: A] :
      ( ( nil(A) = replicate(A,Nb,Xa) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_2695_zip__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Us: list(B),Ys: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Us) )
     => ( zip(A,B,append(A,Xs,Ys),append(B,Us,Vs)) = append(product_prod(A,B),zip(A,B,Xs,Us),zip(A,B,Ys,Vs)) ) ) ).

% zip_append
tff(fact_2696_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(set(A),list(A),linord4507533701916653071of_set(A),bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_2697_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A2: A] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_2698_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M,append(A,Xs,Zs),Ys) = map_upds(A,B,M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_2699_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M,Xs,append(B,Ys,Zs)) = map_upds(A,B,M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_2700_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_2701_list__assn__aux__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),L4: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,nil(A)),L4) = pure_assn(L4 = nil(B)) ).

% list_assn_aux_simps(1)
tff(fact_2702_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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gm(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(aa(set(A),$o,member(A,A2),S3),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_2703_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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gn(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(aa(set(A),$o,member(A,A2),S3),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_2704_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gl(fun(B,A),fun(B,A),F)),A3)) ) ).

% sum_abs
tff(fact_2705_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(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_2706_nth__append__first,axiom,
    ! [A: $tType,I: nat,L: list(A),L4: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,append(A,L,L4)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_append_first
tff(fact_2707_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xa: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,aa(list(A),list(A),cons(A,Xa),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xa ).

% nth_append_length
tff(fact_2708_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] : aa(nat,A,nth(A,append(A,Xs,Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)) = aa(nat,A,nth(A,Ys),Nb) ).

% nth_append_length_plus
tff(fact_2709_drop__eq__Nil2,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,Nb,Xs) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% drop_eq_Nil2
tff(fact_2710_drop__eq__Nil,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( drop(A,Nb,Xs) = nil(A) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% drop_eq_Nil
tff(fact_2711_drop__all,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)
     => ( drop(A,Nb,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_2712_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xa: A,Ys: list(A),Y: A] : list_update(A,append(A,Xs,aa(list(A),list(A),cons(A,Xa),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = append(A,Xs,aa(list(A),list(A),cons(A,Y),Ys)) ).

% list_update_length
tff(fact_2713_take__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : take(A,Nb,append(A,Xs,Ys)) = append(A,take(A,Nb,Xs),take(A,aa(nat,nat,minus_minus(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% take_append
tff(fact_2714_drop__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : drop(A,Nb,append(A,Xs,Ys)) = append(A,drop(A,Nb,Xs),drop(A,aa(nat,nat,minus_minus(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% drop_append
tff(fact_2715_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)
         => ( ( aa(set(A),list(A),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_2716_list__assn__aux__append2,axiom,
    ! [A: $tType,B: $tType,L22: list(A),L24: list(B),P: fun(A,fun(B,assn)),L1: list(A),L13: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L22) = aa(list(B),nat,size_size(list(B)),L24) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,append(A,L1,L22)),append(B,L13,L24)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L1),L13)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L22),L24)) ) ) ).

% list_assn_aux_append2
tff(fact_2717_list__assn__aux__append,axiom,
    ! [A: $tType,B: $tType,L1: list(A),L13: list(B),P: fun(A,fun(B,assn)),L22: list(A),L24: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L13) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,append(A,L1,L22)),append(B,L13,L24)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L1),L13)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L22),L24)) ) ) ).

% list_assn_aux_append
tff(fact_2718_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_gl(fun(B,A),fun(B,A),F)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_2719_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_2720_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_2721_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : concat(A,map(B,list(A),aTP_Lamp_go(fun(B,A),fun(B,list(A)),F),Xs)) = map(B,A,F,Xs) ).

% concat_map_singleton
tff(fact_2722_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.op_ivl_Suc
tff(fact_2723_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gp(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A3)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2724_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))
     => ( map_upds(A,B,M,append(A,Xs,aa(list(A),list(A),cons(A,Xa),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys),Xa,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_2725_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,Xs5: list(A),Y3: A,Ys4: list(A)] :
            ( ( X != Y3 )
            & ( Xs = append(A,Pre,append(A,aa(list(A),list(A),cons(A,X),nil(A)),Xs5)) )
            & ( Ys = append(A,Pre,append(A,aa(list(A),list(A),cons(A,Y3),nil(A)),Ys4)) ) ) ) ) ).

% same_length_different
tff(fact_2726_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aTP_Lamp_gq(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B4)),A3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(set(B),fun(C,A),aTP_Lamp_gs(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B4) ) ).

% sum.swap
tff(fact_2727_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( aa(set(B),$o,member(B,A4),A3)
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_2728_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => ( aa(A,B,G,X) = zero_zero(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_2729_map__by__foldl,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),L: list(B)] : foldl(list(A),B,aTP_Lamp_gt(fun(B,A),fun(list(A),fun(B,list(A))),F),nil(A),L) = map(B,A,F,L) ).

% map_by_foldl
tff(fact_2730_subset__eq__mset__impl_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(list(A),list(A))] :
      ( ! [Ys5: list(A)] : Xa != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys5)
     => ~ ! [X: A,Xs2: list(A),Ys5: list(A)] : Xa != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs2)),Ys5) ) ).

% subset_eq_mset_impl.cases
tff(fact_2731_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K4: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),K4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),aa(A,B,G,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),K4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),K4)) ) ) ).

% sum_mono
tff(fact_2732_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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gu(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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),A3)) ) ).

% sum.distrib
tff(fact_2733_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: 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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),B4)) = aa(set(B),A,aa(fun(B,A),fun(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_gw(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B4)),A3) ) ).

% sum_product
tff(fact_2734_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_gx(fun(B,A),fun(A,fun(B,A)),F),R2)),A3) ) ).

% sum_distrib_right
tff(fact_2735_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,F: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gy(A,fun(fun(B,A),fun(B,A)),R2),F)),A3) ) ).

% sum_distrib_left
tff(fact_2736_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gz(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A3) = aa(A,A,minus_minus(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)) ) ).

% sum_subtractf
tff(fact_2737_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ha(fun(B,A),fun(B,A),F)),A3) = uminus_uminus(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3)) ) ).

% sum_negf
tff(fact_2738_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B4: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hc(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B4),G),R)),A3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_he(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R)),B4) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2739_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 = 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_2740_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 = 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_2741_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,aa(list(A),list(A),cons(A,Xa),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_2742_length__compl__rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [L2: list(A),E: A] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L2))
               => aa(list(A),$o,P,Ll) )
           => aa(list(A),$o,P,append(A,L2,aa(list(A),list(A),cons(A,E),nil(A)))) )
       => aa(list(A),$o,P,L) ) ) ).

% length_compl_rev_induct
tff(fact_2743_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,Nb: nat,Xa: A] : replicate(A,aa(nat,nat,suc,Nb),Xa) = append(A,replicate(A,Nb,Xa),aa(list(A),list(A),cons(A,Xa),nil(A))) ).

% replicate_Suc_conv_snoc
tff(fact_2744_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( append(A,Xs,Ys) = append(A,Ys,Xs) )
         => ? [N: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs2)) = append(A,Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_2745_enumerate__append__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : enumerate(A,Nb,append(A,Xs,Ys)) = append(product_prod(nat,A),enumerate(A,Nb,Xs),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_2746_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_2747_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_2748_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)) ) ) ).

% sum_nonneg
tff(fact_2749_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F: fun(B,A),I5: set(B),G: fun(B,A),I: B] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),I5) )
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,I2)),aa(B,A,G,I2)) )
           => ( aa(set(B),$o,member(B,I),I5)
             => ( aa(set(B),$o,finite_finite2(B),I5)
               => ( aa(B,A,F,I) = aa(B,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_2750_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F: fun(nat,A),G: fun(nat,A)] :
          ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A3)
         => ( ! [X: nat] :
                ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,X)),A3)
               => ( aa(nat,A,F,aa(nat,nat,suc,X)) = aa(nat,A,G,aa(nat,nat,suc,X)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_2751_sorted__append__bigger,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) )
           => sorted_wrt(A,ord_less_eq(A),append(A,Xs,aa(list(A),list(A),cons(A,Y),nil(A)))) ) ) ) ).

% sorted_append_bigger
tff(fact_2752_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(A),Zs: list(B)] : zip(A,B,append(A,Xs,Ys),Zs) = append(product_prod(A,B),zip(A,B,Xs,take(B,aa(list(A),nat,size_size(list(A)),Xs),Zs)),zip(A,B,Ys,drop(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))) ).

% zip_append1
tff(fact_2753_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(B)] : zip(A,B,Xs,append(B,Ys,Zs)) = append(product_prod(A,B),zip(A,B,take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Ys),zip(A,B,drop(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Zs)) ).

% zip_append2
tff(fact_2754_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R12: list(A),R23: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1))
     => ( ( append(A,S1,R12) = append(A,S22,R23) )
       => ? [R1p: list(A)] : R23 = append(A,R1p,R12) ) ) ).

% list_rest_coinc
tff(fact_2755_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),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_hf(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_2756_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_2757_len__greater__imp__nonempty,axiom,
    ! [A: $tType,Xa: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(list(A),nat,size_size(list(A)),L))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_2758_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),Y3: B,Ys5: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys5) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ys5)
               => 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,Y3),Ys5)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_2759_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),$o)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X: A,Xs2: list(A),Y3: B,Ys5: list(B),Z3: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys5) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs2),Ys5),Zs2)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,aa(list(A),list(A),cons(A,X),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys5)),aa(list(C),list(C),cons(C,Z3),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs),Ys),Zs) ) ) ) ) ).

% list_induct3
tff(fact_2760_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws: list(D6),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D6),$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)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = 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),Y3: B,Ys5: list(B),Z3: 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)),Ys5) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = 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),Ys5),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,Y3),Ys5)),aa(list(C),list(C),cons(C,Z3),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),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_2761_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_2762_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S3) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_hh(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S3),H),G)),aa(set(A),set(C),image(A,C,G),S3)) ) ) ) ).

% sum.image_gen
tff(fact_2763_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_2764_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_2765_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_2766_replicate__0,axiom,
    ! [A: $tType,Xa: A] : replicate(A,zero_zero(nat),Xa) = nil(A) ).

% replicate_0
tff(fact_2767_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_2768_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,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hj(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_2769_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,nat)] : size_list(A,Xa,nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_2770_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3) = zero_zero(B) )
            <=> ! [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                 => ( aa(A,B,F,X3) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2771_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S2: set(A),Ta: set(B),G: fun(B,C),I: fun(B,A),F: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),Ta)
           => ( ! [X: B] :
                  ( aa(set(B),$o,member(B,X),Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X)) )
             => ( ! [X: A] :
                    ( aa(set(A),$o,member(A,X),S2)
                   => ? [Xa2: B] :
                        ( aa(set(B),$o,member(B,Xa2),Ta)
                        & ( aa(B,A,I,Xa2) = X )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F,X)),aa(B,C,G,Xa2)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F),S2)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_2772_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) )
           => ( ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X4)),aa(A,B,G,X4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_2773_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),A2),Xs) = append(A,Xs,aa(list(A),list(A),cons(A,A2),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_2774_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X15: A,Y1: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X15),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S3)
             => ( ! [X: B] :
                    ( aa(set(B),$o,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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3)) ) ) ) ) ) ).

% sum.related
tff(fact_2775_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,G,X)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum_strict_mono
tff(fact_2776_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,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),set_or7035219750837199246ssThan(A,C2,D)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_2777_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S6: set(A),T5: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T2: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S6)
         => ( aa(set(B),$o,finite_finite2(B),T5)
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),S3),S6))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),S3),S6))
                   => aa(set(B),$o,member(B,aa(A,B,J,A4)),aa(set(B),set(B),minus_minus(set(B),T2),T5)) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),minus_minus(set(B),T2),T5))
                     => ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),minus_minus(set(B),T2),T5))
                       => aa(set(A),$o,member(A,aa(B,A,I,B3)),aa(set(A),set(A),minus_minus(set(A),S3),S6)) )
                   => ( ! [A4: A] :
                          ( aa(set(A),$o,member(A,A4),S6)
                         => ( aa(A,C,G,A4) = zero_zero(C) ) )
                     => ( ! [B3: B] :
                            ( aa(set(B),$o,member(B,B3),T5)
                           => ( aa(B,C,H,B3) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( aa(set(A),$o,member(A,A4),S3)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),G),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),T2) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_2778_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_2779_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,P3: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P3)
           => ( aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,M,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,M,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,Nb,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_2780_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,aa(nat,nat,suc,I),Xs) = append(A,take(A,I,Xs),aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_2781_take__update__last,axiom,
    ! [A: $tType,Nb: nat,List: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),List))
     => ( list_update(A,take(A,aa(nat,nat,suc,Nb),List),Nb,Xa) = append(A,take(A,Nb,List),aa(list(A),list(A),cons(A,Xa),nil(A))) ) ) ).

% take_update_last
tff(fact_2782_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),append(A,Xs,Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X3: A] :
                ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Xs))
               => ! [Xa3: A] :
                    ( aa(set(A),$o,member(A,Xa3),aa(list(A),set(A),set2(A),Ys))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa3) ) ) ) ) ) ).

% sorted_append
tff(fact_2783_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),nil(A),A3) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_2784_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S2: set(A),F: fun(A,B),B4: B,I: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),S2) = B4 )
             => ( aa(set(A),$o,member(A,I),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),B4) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2785_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S2: set(A),F: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),S2) = zero_zero(B) )
             => ( aa(set(A),$o,member(A,I),S2)
               => ( aa(A,B,F,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2786_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,append(A,Xs,Ys),I,Xa) = append(A,list_update(A,Xs,I,Xa),Ys) ) ) ).

% list_update_append1
tff(fact_2787_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S3: set(A),T2: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S3)),T2)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S3),G),H)),T2) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),H),S3) ) ) ) ) ) ).

% sum.group
tff(fact_2788_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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_hm(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2789_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,I5: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hn(A,fun(nat,fun(nat,A)),Xa),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),I5)) ) ).

% sum_power_add
tff(fact_2790_drop__take__drop__unsplit,axiom,
    ! [A: $tType,I: nat,J: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( append(A,drop(A,I,take(A,J,L)),drop(A,J,L)) = drop(A,I,L) ) ) ).

% drop_take_drop_unsplit
tff(fact_2791_append__eq__conv__conj,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( append(A,Xs,Ys) = Zs )
    <=> ( ( Xs = take(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) )
        & ( Ys = drop(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_2792_length__compl__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [E: A,L2: list(A)] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L2))
               => aa(list(A),$o,P,Ll) )
           => aa(list(A),$o,P,aa(list(A),list(A),cons(A,E),L2)) )
       => aa(list(A),$o,P,L) ) ) ).

% length_compl_induct
tff(fact_2793_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A4: A] : L = aa(list(A),list(A),cons(A,A4),nil(A)) ) ).

% list_decomp_1
tff(fact_2794_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xa),nil(A))) ) ).

% sorted1
tff(fact_2795_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_cj(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_cj(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_2796_merge_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Xaa: list(A),Y: list(A)] :
          ( ( merge(A,Xa,Xaa) = Y )
         => ( ( ( Xa = nil(A) )
             => ( Y != Xaa ) )
           => ( ! [V2: A,Va: list(A)] :
                  ( ( Xa = aa(list(A),list(A),cons(A,V2),Va) )
                 => ( ( Xaa = nil(A) )
                   => ( Y != aa(list(A),list(A),cons(A,V2),Va) ) ) )
             => ~ ! [X15: A,L12: list(A)] :
                    ( ( Xa = aa(list(A),list(A),cons(A,X15),L12) )
                   => ! [X23: A,L23: list(A)] :
                        ( ( Xaa = aa(list(A),list(A),cons(A,X23),L23) )
                       => ( Y != $ite(
                              aa(A,$o,aa(A,fun(A,$o),ord_less(A),X15),X23),
                              aa(list(A),list(A),cons(A,X15),merge(A,L12,aa(list(A),list(A),cons(A,X23),L23))),
                              $ite(X15 = X23,aa(list(A),list(A),cons(A,X15),merge(A,L12,L23)),aa(list(A),list(A),cons(A,X23),merge(A,aa(list(A),list(A),cons(A,X15),L12),L23))) ) ) ) ) ) ) ) ) ).

% merge.elims
tff(fact_2797_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( aa(set(A),$o,member(A,I),I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F,I))
             => ( ! [I2: A] :
                    ( aa(set(A),$o,member(A,I2),I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_2798_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),I5)) ) ) ) ) ).

% sum_pos
tff(fact_2799_list__assn__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A2: A,As: list(A)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),cons(A,A2),As)),nil(B)) = bot_bot(assn) ).

% list_assn_simps(3)
tff(fact_2800_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_2801_list__assn_Osimps_I3_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Va2: A,Vaa: list(A)] : aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,aa(list(A),list(A),cons(A,Va2),Vaa)),nil(B)) = bot_bot(assn) ).

% list_assn.simps(3)
tff(fact_2802_list__assn_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Va2: B,Vaa: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,nil(A)),aa(list(B),list(B),cons(B,Va2),Vaa)) = bot_bot(assn) ).

% list_assn.simps(4)
tff(fact_2803_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( aa(set(A),$o,member(A,X),S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2804_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,H,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( aa(set(A),$o,member(A,X),S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2805_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2806_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T2) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2807_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),minus_minus(set(A),C3),B4))
                     => ( aa(A,B,H,B3) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C3) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2808_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),minus_minus(set(A),C3),B4))
                     => ( aa(A,B,H,B3) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B4) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C3) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2809_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,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2810_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),minus_minus(set(A),A3),B4)) = aa(B,B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),B4)) ) ) ) ) ).

% sum_diff
tff(fact_2811_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_2812_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(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_2813_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_2814_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_2815_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] :
      aa(nat,A,nth(A,append(A,Xs,Ys)),Nb) = $ite(aa(nat,$o,aa(nat,fun(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_2816_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      subseqs(A,aa(list(A),list(A),cons(A,Xa),Xs)) = $let(
        xss: list(list(A)),
        xss:= subseqs(A,Xs),
        append(list(A),map(list(A),list(A),cons(A,Xa),xss),xss) ) ).

% subseqs.simps(2)
tff(fact_2817_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat,Xa: A] :
      list_update(A,append(A,Xs,Ys),Nb,Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),append(A,list_update(A,Xs,Nb,Xa),Ys),append(A,Xs,list_update(A,Ys,aa(nat,nat,minus_minus(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)),Xa))) ).

% list_update_append
tff(fact_2818_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( append(A,Xs_1,Xs_2) = append(A,Ys_1,Ys_2) )
    <=> $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)),
          ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
          & ( Xs_2 = append(A,drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1),Ys_2) ) ),
          ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
          & ( append(A,drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1),Xs_2) = Ys_2 ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_2819_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ho(fun(nat,A),fun(nat,A),F)),set_or7035219750837199246ssThan(nat,M,Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F,Nb)),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_2820_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or7035219750837199246ssThan(nat,Nb,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_2821_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B4: set(A),A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => ( ! [B3: A] :
                  ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),minus_minus(set(A),B4),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,B3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),B4)) ) ) ) ) ).

% sum_mono2
tff(fact_2822_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).

% last_zip
tff(fact_2823_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,Xa),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_2824_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),insert(A,Xa),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_2825_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),F: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(B,B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(A,B,F,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)) ) ) ) ).

% sum_diff1
tff(fact_2826_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
       => $let(
            p: nat,
            p:= aa(list(A),nat,size_size(list(A)),Ys),
            slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),p),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),p),append(A,Ys,Xs)) ) ) ) ).

% slice_prepend
tff(fact_2827_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,aa(fun(A,B),fun(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_hq(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S3) = $ite(aa(set(A),$o,member(A,A2),S3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_2828_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),append(B,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,groups4207007520872428315er_sum(B,A,F,A2),Ys))) ) ).

% horner_sum_append
tff(fact_2829_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A3: set(A),B2: A,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),minus_minus(set(A),B4),A3))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F,B2))
               => ( ! [X: A] :
                      ( aa(set(A),$o,member(A,X),B4)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),B4)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2830_take__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] :
      take(A,Nb,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(Nb = zero_zero(nat),nil(A),aa(list(A),list(A),cons(A,Xa),take(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xs))) ).

% take_Cons'
tff(fact_2831_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X)) )
           => ( aa(set(A),$o,finite_finite2(A),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_2832_last__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( last(A,Xs) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_2833_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xa: A] :
      ( ( Xs != nil(A) )
     => ( last(A,list_update(A,Xs,K,Xa)) = $ite(K = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xa,last(A,Xs)) ) ) ).

% last_list_update
tff(fact_2834_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xa: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xa,I2)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xa),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,A2,I2)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_hr(fun(A,B),fun(fun(A,B),fun(A,B)),Xa),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2835_list__assn_Oelims,axiom,
    ! [B: $tType,A: $tType,Xa: fun(A,fun(B,assn)),Xaa: list(A),Xb: list(B),Y: assn] :
      ( ( aa(list(B),assn,vEBT_List_list_assn(A,B,Xa,Xaa),Xb) = Y )
     => ( ( ( Xaa = nil(A) )
         => ( ( Xb = nil(B) )
           => ( Y != one_one(assn) ) ) )
       => ( ! [A4: A,As2: list(A)] :
              ( ( Xaa = aa(list(A),list(A),cons(A,A4),As2) )
             => ! [C5: B,Cs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),cons(B,C5),Cs2) )
                 => ( Y != aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Xa,A4),C5)),aa(list(B),assn,vEBT_List_list_assn(A,B,Xa,As2),Cs2)) ) ) )
         => ( ( ? [V2: A,Va: list(A)] : Xaa = aa(list(A),list(A),cons(A,V2),Va)
             => ( ( Xb = nil(B) )
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ( Xaa = nil(A) )
               => ( ? [V2: B,Va: list(B)] : Xb = aa(list(B),list(B),cons(B,V2),Va)
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_2836_id__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( Xs = append(A,take(A,I,Xs),aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_2837_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I2)),aa(nat,A,A2,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I2)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_hs(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_2838_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arcosh(A,one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_2839_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I2)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I2)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_ht(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,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_2840_list__assn_Opelims,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,fun(B,assn)),Xaa: list(A),Xb: list(B),Y: assn] :
      ( ( aa(list(B),assn,vEBT_List_list_assn(A,B,Xa,Xaa),Xb) = Y )
     => ( aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xaa),Xb)))
       => ( ( ( Xaa = nil(A) )
           => ( ( Xb = nil(B) )
             => ( ( Y = one_one(assn) )
               => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) ) )
         => ( ! [A4: A,As2: list(A)] :
                ( ( Xaa = aa(list(A),list(A),cons(A,A4),As2) )
               => ! [C5: B,Cs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),cons(B,C5),Cs2) )
                   => ( ( Y = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Xa,A4),C5)),aa(list(B),assn,vEBT_List_list_assn(A,B,Xa,As2),Cs2)) )
                     => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,A4),As2)),aa(list(B),list(B),cons(B,C5),Cs2)))) ) ) )
           => ( ! [V2: A,Va: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),cons(A,V2),Va) )
                 => ( ( Xb = nil(B) )
                   => ( ( Y = bot_bot(assn) )
                     => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,V2),Va)),nil(B)))) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ! [V2: B,Va: list(B)] :
                      ( ( Xb = aa(list(B),list(B),cons(B,V2),Va) )
                     => ( ( Y = bot_bot(assn) )
                       => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xa),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),cons(B,V2),Va)))) ) ) ) ) ) ) ) ) ).

% list_assn.pelims
tff(fact_2841_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( artanh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_2842_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arsinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_2843_int__sum,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7311177749621191930dd_sum(A,int),aTP_Lamp_hu(fun(A,nat),fun(A,int),F)),A3) ).

% int_sum
tff(fact_2844_sum__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),multiset(B),aa(fun(A,multiset(B)),fun(set(A),multiset(B)),groups7311177749621191930dd_sum(A,multiset(B)),F),A3) = zero_zero(multiset(B)) )
      <=> ! [X3: A] :
            ( aa(set(A),$o,member(A,X3),A3)
           => ( aa(A,multiset(B),F,X3) = zero_zero(multiset(B)) ) ) ) ) ).

% sum_eq_empty_iff
tff(fact_2845_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F: fun(A,nat)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X)),aa(A,nat,F,X)) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_hv(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F)),A3) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_2846_sum__SucD,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A),Nb: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3) = aa(nat,nat,suc,Nb) )
     => ? [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F,X)) ) ) ).

% sum_SucD
tff(fact_2847_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X3: A] :
            ( aa(set(A),$o,member(A,X3),A3)
            & ( aa(A,nat,F,X3) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( aa(set(A),$o,member(A,Xa3),A3)
               => ( ( X3 != Xa3 )
                 => ( aa(A,nat,F,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2848_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3) = one_one(nat) )
      <=> ? [X3: A] :
            ( aa(set(A),$o,member(A,X3),A3)
            & ( aa(A,nat,F,X3) = one_one(nat) )
            & ! [Xa3: A] :
                ( aa(set(A),$o,member(A,Xa3),A3)
               => ( ( X3 != Xa3 )
                 => ( aa(A,nat,F,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2849_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_hw(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_hx(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2850_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_hw(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_gg(nat,fun(complex,fun(complex,$o)),Nb),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2851_sum__diff__nat,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),minus_minus(set(A),A3),B4)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_2852_sum__diff1__nat,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A),A2: A] :
      aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3)),aa(A,nat,F,A2)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3)) ).

% sum_diff1_nat
tff(fact_2853_merge_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Xaa: list(A),Y: list(A)] :
          ( ( merge(A,Xa,Xaa) = Y )
         => ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xaa))
           => ( ( ( Xa = nil(A) )
               => ( ( Y = Xaa )
                 => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xaa)) ) )
             => ( ! [V2: A,Va: list(A)] :
                    ( ( Xa = aa(list(A),list(A),cons(A,V2),Va) )
                   => ( ( Xaa = nil(A) )
                     => ( ( Y = aa(list(A),list(A),cons(A,V2),Va) )
                       => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,V2),Va)),nil(A))) ) ) )
               => ~ ! [X15: A,L12: list(A)] :
                      ( ( Xa = aa(list(A),list(A),cons(A,X15),L12) )
                     => ! [X23: A,L23: list(A)] :
                          ( ( Xaa = aa(list(A),list(A),cons(A,X23),L23) )
                         => ( ( Y = $ite(
                                  aa(A,$o,aa(A,fun(A,$o),ord_less(A),X15),X23),
                                  aa(list(A),list(A),cons(A,X15),merge(A,L12,aa(list(A),list(A),cons(A,X23),L23))),
                                  $ite(X15 = X23,aa(list(A),list(A),cons(A,X15),merge(A,L12,L23)),aa(list(A),list(A),cons(A,X23),merge(A,aa(list(A),list(A),cons(A,X15),L12),L23))) ) )
                           => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X15),L12)),aa(list(A),list(A),cons(A,X23),L23))) ) ) ) ) ) ) ) ) ).

% merge.pelims
tff(fact_2854_merge__list__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ls: list(list(A)),As: list(list(A))] :
          ( ! [L2: list(A)] :
              ( aa(set(list(A)),$o,member(list(A),L2),aa(list(list(A)),set(list(A)),set2(list(A)),Ls))
             => ( distinct(A,L2)
                & sorted_wrt(A,ord_less_eq(A),L2) ) )
         => ( ! [L2: list(A)] :
                ( aa(set(list(A)),$o,member(list(A),L2),aa(list(list(A)),set(list(A)),set2(list(A)),As))
               => ( distinct(A,L2)
                  & sorted_wrt(A,ord_less_eq(A),L2) ) )
           => ( distinct(A,merge_list(A,As,Ls))
              & sorted_wrt(A,ord_less_eq(A),merge_list(A,As,Ls))
              & ( aa(list(A),set(A),set2(A),merge_list(A,As,Ls)) = aa(list(A),set(A),set2(A),concat(A,append(list(A),As,Ls))) ) ) ) ) ) ).

% merge_list_correct
tff(fact_2855_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X5) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_2856_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F: fun(nat,A),Ns: list(nat)] :
          ( ! [X: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,X)),aa(nat,A,F,Y3)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,map(nat,A,F,Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_2857_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image(A,list(A),aTP_Lamp_hy(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_2858_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_2859_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Ns))
             => ( X3 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_2860_count__notin,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( ~ aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xa) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_2861_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : groups8242544230860333062m_list(A,map(B,A,aTP_Lamp_hz(B,A),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_2862_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Xs: list(A)] :
          ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),groups8242544230860333062m_list(A,Xs)) ) ) ).

% member_le_sum_list
tff(fact_2863_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gu(fun(B,A),fun(fun(B,A),fun(B,A)),F),G),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,map(B,A,F,Xs))),groups8242544230860333062m_list(A,map(B,A,G,Xs))) ) ).

% sum_list_addf
tff(fact_2864_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gy(A,fun(fun(B,A),fun(B,A)),C2),F),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,map(B,A,F,Xs))) ) ).

% sum_list_const_mult
tff(fact_2865_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),C2: A,Xs: list(B)] : groups8242544230860333062m_list(A,map(B,A,aa(A,fun(B,A),aTP_Lamp_gx(fun(B,A),fun(A,fun(B,A)),F),C2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,map(B,A,F,Xs))),C2) ) ).

% sum_list_mult_const
tff(fact_2866_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gz(fun(B,A),fun(fun(B,A),fun(B,A)),F),G),Xs)) = aa(A,A,minus_minus(A,groups8242544230860333062m_list(A,map(B,A,F,Xs))),groups8242544230860333062m_list(A,map(B,A,G,Xs))) ) ).

% sum_list_subtractf
tff(fact_2867_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,map(A,nat,F,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_ia(fun(A,nat),fun(list(A),fun(A,nat)),F),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_2868_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_2869_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) )
         => ( ( groups8242544230860333062m_list(A,Xs) = zero_zero(A) )
          <=> ! [X3: A] :
                ( aa(set(A),$o,member(A,X3),aa(list(A),set(A),set2(A),Xs))
               => ( X3 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_2870_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),groups8242544230860333062m_list(A,Xs)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_2871_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs))),groups8242544230860333062m_list(A,map(A,A,abs_abs(A),Xs))) ) ).

% sum_list_abs
tff(fact_2872_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = foldr(A,A,plus_plus(A),Xs,zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_2873_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat,C2: A] : groups8242544230860333062m_list(A,replicate(A,Nb,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),C2) ) ).

% sum_list_replicate
tff(fact_2874_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,map(A,B,F,Xs))),groups8242544230860333062m_list(B,map(A,B,G,Xs))) ) ) ).

% sum_list_mono
tff(fact_2875_length__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,Xss)) = groups8242544230860333062m_list(nat,map(list(A),nat,size_size(list(A)),Xss)) ).

% length_concat
tff(fact_2876_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(A,Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ib(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_2877_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_2878_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),Xa)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% count_le_length
tff(fact_2879_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),groups8242544230860333062m_list(A,Ns)) ) ) ).

% elem_le_sum_list
tff(fact_2880_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,G,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,map(A,B,F,Xs))),groups8242544230860333062m_list(B,map(A,B,G,Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_2881_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( groups8242544230860333062m_list(nat,map(A,nat,F,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_ic(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F)),X5) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_2882_size__list__conv__sum__list,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] : size_list(A,F,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,map(A,nat,F,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% size_list_conv_sum_list
tff(fact_2883_sum__list__Suc,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,map(A,nat,aTP_Lamp_id(fun(A,nat),fun(A,nat),F),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,map(A,nat,F,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_2884_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R2: A,Xs: list(B)] : groups8242544230860333062m_list(A,map(B,A,aTP_Lamp_ie(A,fun(B,A),R2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs))),R2) ) ).

% sum_list_triv
tff(fact_2885_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_2886_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs,K,Xa)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),Xa)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_2887_mergesort__remdups__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : mergesort_remdups(A,Xs) = merge_list(A,nil(list(A)),map(A,list(A),aTP_Lamp_if(A,list(A)),Xs)) ) ).

% mergesort_remdups_def
tff(fact_2888_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_2889_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,Z: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb) = A2 )
          <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ig(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_2890_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( ln_ln(A,one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2891_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2))
     => ~ ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys) )
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys),I2))),R2) ) ) ) ) ).

% lex_take_index
tff(fact_2892_ln__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ( ln_ln(real,Xa) = ln_ln(real,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2893_ln__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),ln_ln(real,Xa)),ln_ln(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2894_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( aa(set(A),$o,member(A,I),aa(A,set(A),set_ord_atMost(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K) ) ) ).

% atMost_iff
tff(fact_2895_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K))) = append(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,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_2896_ln__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,Xa)),ln_ln(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2897_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),Xa)),aa(A,set(A),set_ord_atMost(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% atMost_subset_iff
tff(fact_2898_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( ln_ln(real,Xa) = zero_zero(real) )
      <=> ( Xa = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_2899_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),ln_ln(real,Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_2900_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),ln_ln(real,Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_2901_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),ln_ln(real,Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_2902_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_2903_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atMost_Suc
tff(fact_2904_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_2905_Cons__in__lex,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,Xa),Xs)),aa(list(A),list(A),cons(A,Y),Ys))),lex(A,R2))
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y)),R2)
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xa = Y )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2)) ) ) ) ).

% Cons_in_lex
tff(fact_2906_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,fun(list(C),B)),List: list(C)] : aa(B,A,H,aa(list(C),B,case_list(B,C,F1,F22),List)) = aa(list(C),A,case_list(A,C,aa(B,A,H,F1),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_ih(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H),F22)),List) ).

% list.case_distrib
tff(fact_2907_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_2908_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ii(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_2909_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),ln_ln(real,Xa)),Xa) ) ).

% ln_less_self
tff(fact_2910_atMost__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% atMost_Suc
tff(fact_2911_finite__nat__iff__bounded__le,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S3)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S3),aa(nat,set(nat),set_ord_atMost(nat),K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_2912_ln__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,Xa)),Xa) ) ).

% ln_bound
tff(fact_2913_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),ln_ln(real,Xa)) ) ).

% ln_gt_zero
tff(fact_2914_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),ln_ln(real,Xa)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_2915_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),ln_ln(real,Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_2916_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),ln_ln(real,Xa)) ) ).

% ln_ge_zero
tff(fact_2917_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),ln_ln(real,Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2918_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_2919_ln__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ln_ln(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),ln_ln(real,Xa)),ln_ln(real,Y)) ) ) ) ).

% ln_mult
tff(fact_2920_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( ln_ln(real,Xa) = aa(real,real,minus_minus(real,Xa),one_one(real)) )
       => ( Xa = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_2921_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_2922_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),I: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ij(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_atMost(nat),I)) = aa(A,A,minus_minus(A,aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_2923_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,D: fun(nat,A)] :
          ( ! [X3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),D),X3)),aa(nat,set(nat),set_ord_atMost(nat),Nb))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
             => ( aa(nat,A,C2,I3) = aa(nat,A,D,I3) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_2924_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_2925_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Us)),append(A,Ys,Vs))),lex(A,R2)) ) ) ).

% lex_append_rightI
tff(fact_2926_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,Xa)),aa(real,real,minus_minus(real,Xa),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_2927_ln__add__one__self__le__self2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self2
tff(fact_2928_ln__realpow,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ln_ln(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),ln_ln(real,Xa)) ) ) ).

% ln_realpow
tff(fact_2929_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_2930_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( ! [X3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
             => ( aa(nat,A,C2,I3) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_2931_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,aa(real,real,minus_minus(real,one_one(real)),Xa))),uminus_uminus(real,Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2932_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),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),Xa),aa(nat,nat,suc,Nb))) ) ).

% sum_gp_basic
tff(fact_2933_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_im(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_2934_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_im(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
        <=> ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,A,C2,I3) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_2935_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xa: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),J2)
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),B2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_ip(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_2936_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xa: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
         => ( aa(nat,nat,A2,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),J2)
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_iq(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_iq(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_is(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_2937_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,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(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_it(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(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_iu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2938_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),$o)] :
      ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),A0)
     => ( ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),nil(list(A)))
         => aa(list(list(A)),$o,P,nil(list(A))) )
       => ( ! [Xss2: list(list(A))] :
              ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2))
             => ( aa(list(list(A)),$o,P,Xss2)
               => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) ) )
         => ( ! [X: A,Xs2: list(A),Xss2: list(list(A))] :
                ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss2))
               => ( aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_iv(A,fun(list(A),list(list(A))))),Xss2))))
                 => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss2)) ) )
           => aa(list(list(A)),$o,P,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_2939_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,Xa: 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,Xa),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xa),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_iw(A,fun(list(A),list(A)))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_iv(A,fun(list(A),list(list(A))))),Xss))))) ).

% transpose.simps(3)
tff(fact_2940_transpose_Oelims,axiom,
    ! [A: $tType,Xa: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xa) = Y )
     => ( ( ( Xa = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( Xa = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( Xa = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_iw(A,fun(list(A),list(A)))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_iv(A,fun(list(A),list(list(A))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_2941_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),Nb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ? [M10: real] :
            ! [Z4: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M10),real_V7770717601297561774m_norm(A,Z4))
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ix(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),aa(nat,set(nat),set_ord_atMost(nat),Nb)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_2942_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Nb)))),aa(A,A,minus_minus(A,one_one(A)),Xa))) ) ).

% sum_gp0
tff(fact_2943_div__of__0__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),zero_zero(word(A))),Nb) = zero_zero(word(A)) ) ).

% div_of_0_id
tff(fact_2944_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_2945_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% div_0
tff(fact_2946_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_2947_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_2948_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_2949_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_2950_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% div_by_1
tff(fact_2951_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_2952_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_2953_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_2954_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ).

% times_divide_eq_left
tff(fact_2955_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_2956_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_2957_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_2958_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_2959_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_2960_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_2961_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_2962_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_2963_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_2964_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_2965_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2966_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2967_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2968_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2969_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2970_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2971_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2972_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( ( real_V7770717601297561774m_norm(A,Xa) = zero_zero(real) )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2973_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2974_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,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_2975_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_2976_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_2977_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_2978_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_2979_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_2980_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_2981_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_2982_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_2983_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2984_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2985_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2986_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2987_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xa))
        <=> ( Xa != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2988_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xa)),zero_zero(real))
        <=> ( Xa = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2989_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_2990_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_2991_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_2992_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_2993_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_iy(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D)),A3) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A3)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2994_list__encode_Ocases,axiom,
    ! [Xa: list(nat)] :
      ( ( Xa != nil(nat) )
     => ~ ! [X: nat,Xs2: list(nat)] : Xa != aa(list(nat),list(nat),cons(nat,X),Xs2) ) ).

% list_encode.cases
tff(fact_2995_real__of__nat__div4,axiom,
    ! [Nb: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xa))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xa))) ).

% real_of_nat_div4
tff(fact_2996_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2997_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_2998_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_2999_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_3000_div__by__0__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% div_by_0_word
tff(fact_3001_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_iz(fun(B,A),fun(A,fun(B,A)),F),R2)),A3) ) ).

% sum_divide_distrib
tff(fact_3002_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_3003_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_3004_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_3005_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_3006_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_3007_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_3008_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_right_mono
tff(fact_3009_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_3010_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3011_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_3012_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_3013_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_3014_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_3015_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_3016_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_3017_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_3018_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_3019_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_3020_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3021_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3022_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_3023_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_3024_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_3025_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_3026_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_3027_real__of__nat__div2,axiom,
    ! [Nb: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xa)))) ).

% real_of_nat_div2
tff(fact_3028_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y)) ) ).

% norm_mult
tff(fact_3029_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_3030_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xa)) ) ).

% norm_ge_zero
tff(fact_3031_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),uminus_uminus(A,B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_3032_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),uminus_uminus(A,A2)),uminus_uminus(A,B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3033_real__of__nat__div3,axiom,
    ! [Nb: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xa)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_3034_sum__norm__le,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S3: set(A),F: fun(A,B),G: fun(A,real)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),S3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F,X))),aa(A,real,G,X)) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),S3))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),G),S3)) ) ) ).

% sum_norm_le
tff(fact_3035_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_ja(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_3036_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_jb(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_3037_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_3038_complex__mod__minus__le__complex__mod,axiom,
    ! [Xa: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,real_V7770717601297561774m_norm(complex,Xa))),real_V7770717601297561774m_norm(complex,Xa)) ).

% complex_mod_minus_le_complex_mod
tff(fact_3039_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B2),A2))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A2)) ).

% complex_mod_triangle_ineq2
tff(fact_3040_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3041_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3042_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_3043_div__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),W) = one_one(word(A)) ) ) ) ).

% div_word_self
tff(fact_3044_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y) = zero_zero(word(A)) ) ) ) ).

% word_div_lt_eq_0
tff(fact_3045_word__less__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y) = zero_zero(word(A)) )
         => ( ( Y = zero_zero(word(A)) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_less_div
tff(fact_3046_word__div__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Va2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),Va2)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),Va2) = zero_zero(word(A)) ) ) ) ).

% word_div_less
tff(fact_3047_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)),B2)),A2) ) ).

% word_div_mult_le
tff(fact_3048_div__to__mult__word__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Y),Z))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Z)),Y) ) ) ).

% div_to_mult_word_lt
tff(fact_3049_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),aTP_Lamp_jc(fun(B,A),fun(B,real),F)),A3)) ) ).

% norm_sum
tff(fact_3050_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,aa(nat,fun(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_3051_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_positive
tff(fact_3052_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3053_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_3054_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_3055_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_3056_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_3057_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_3058_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less2
tff(fact_3059_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less
tff(fact_3060_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_le
tff(fact_3061_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3062_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_3063_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_3064_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3065_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_3066_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_3067_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_3068_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_3069_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_3070_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_3071_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_3072_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_3073_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_3074_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_3075_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_3076_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_3077_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_3078_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_3079_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_3080_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_3081_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_3082_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2) ) ) ).

% gt_half_sum
tff(fact_3083_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_3084_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_3085_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_3086_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3087_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = uminus_uminus(A,one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = uminus_uminus(A,B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_3088_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = uminus_uminus(A,A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_3089_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = C2 )
          <=> ( uminus_uminus(A,A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_3090_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A2 )
        <=> $ite(C2 != zero_zero(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_3091_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = uminus_uminus(A,B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_3092_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,R2: real,Y: A,S2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S2)) ) ) ) ).

% norm_mult_less
tff(fact_3093_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_3094_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),E2) ) ) ).

% norm_triangle_le
tff(fact_3095_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,S2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S2)) ) ) ) ).

% norm_triangle_mono
tff(fact_3096_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_3097_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),C2)) ) ) ).

% norm_add_leD
tff(fact_3098_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A,Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xa)),Nb)) ) ).

% norm_power_ineq
tff(fact_3099_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Y)))) ) ).

% norm_triangle_sub
tff(fact_3100_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))) ) ).

% norm_triangle_ineq4
tff(fact_3101_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_le
tff(fact_3102_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Y))),E2) ) ) ).

% norm_triangle_le_diff
tff(fact_3103_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),Xa)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% abs_div_pos
tff(fact_3104_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ).

% norm_diff_ineq
tff(fact_3105_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% norm_triangle_ineq2
tff(fact_3106_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_3107_zdiv__mono1,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A6)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A6),B2)) ) ) ).

% zdiv_mono1
tff(fact_3108_zdiv__mono2,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B6)) ) ) ) ).

% zdiv_mono2
tff(fact_3109_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3110_zdiv__mono1__neg,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A6)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A6),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_3111_zdiv__mono2__neg,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B6)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3112_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_3113_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)) ) ) ).

% div_positive_int
tff(fact_3114_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_3115_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_3116_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3117_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3118_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3119_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3120_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),K)),Xa) ) ) ).

% int_div_less_self
tff(fact_3121_ln__div,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ln_ln(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,minus_minus(real,ln_ln(real,Xa)),ln_ln(real,Y)) ) ) ) ).

% ln_div
tff(fact_3122_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( ( Nb != one_one(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Nb)),Xa) ) ) ) ).

% div_less_dividend_word
tff(fact_3123_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
           => aa(word(A),$o,aa(word(A),fun(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),Xa)),K) ) ) ) ).

% div_lt_mult
tff(fact_3124_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),C2)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),C2)),B2) ) ) ) ).

% More_Word.word_div_mult
tff(fact_3125_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_3126_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_3127_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_3128_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_3129_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_3130_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_3131_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_3132_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_3133_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_3134_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_3135_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_3136_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_3137_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_3138_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_3139_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_3140_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(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_3141_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_3142_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_3143_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(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_3144_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_3145_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_3146_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),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_3147_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_3148_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),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_3149_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,uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = $ite(Z = zero_zero(A),uminus_uminus(A,B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(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_3150_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),uminus_uminus(A,B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_3151_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,uminus_uminus(A,Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_3152_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          real_V7770717601297561774m_norm(A,sgn_sgn(A,Xa)) = $ite(Xa = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_3153_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Nb: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,M),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_diff
tff(fact_3154_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% norm_triangle_ineq3
tff(fact_3155_div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_3156_msrevs_I1_J,axiom,
    ! [Nb: nat,K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)),M)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),K) ) ) ).

% msrevs(1)
tff(fact_3157_finite__int__iff__bounded__le,axiom,
    ! [S3: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S3)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S3)),aa(int,set(int),set_ord_atMost(int),K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_3158_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),uminus_uminus(int,Nb))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),Nb)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_3159_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),uminus_uminus(int,one_one(int))),B2) = uminus_uminus(int,one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_3160_ln__diff__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,ln_ln(real,Xa)),ln_ln(real,Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,Xa),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_3161_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xa)),K) ) ) ) ).

% div_le_mult
tff(fact_3162_div__sgn__abs__cancel,axiom,
    ! [Va2: int,K: int,L: int] :
      ( ( Va2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,Va2)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,Va2)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_3163_case__list__rule,axiom,
    ! [A: $tType,B: $tType,L: list(A),P: assn,Fn: heap_Time_Heap(B),Q: fun(B,assn),Fc: fun(A,fun(list(A),heap_Time_Heap(B)))] :
      ( ( ( L = nil(A) )
       => hoare_hoare_triple(B,P,Fn,Q) )
     => ( ! [X: A,Xs2: list(A)] :
            ( ( L = aa(list(A),list(A),cons(A,X),Xs2) )
           => hoare_hoare_triple(B,P,aa(list(A),heap_Time_Heap(B),aa(A,fun(list(A),heap_Time_Heap(B)),Fc,X),Xs2),Q) )
       => hoare_hoare_triple(B,P,aa(list(A),heap_Time_Heap(B),case_list(heap_Time_Heap(B),A,Fn,Fc),L),Q) ) ) ).

% case_list_rule
tff(fact_3164_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Va2: A,R2: A,S2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),Va2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,minus_minus(A,Va2),U))),S2))),Va2) ) ) ) ) ).

% scaling_mono
tff(fact_3165_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_3166_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_3167_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_3168_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_3169_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),uminus_uminus(A,B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_3170_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_3171_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
         => ~ ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E2) ) ) ).

% nat_approx_posE
tff(fact_3172_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N6: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),K6) )
        <=> ? [N8: nat] :
            ! [N6: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% lemma_NBseq_def2
tff(fact_3173_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N6: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),K6) )
        <=> ? [N8: nat] :
            ! [N6: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% lemma_NBseq_def
tff(fact_3174_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( ( Nb != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_3175_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q4: 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),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3176_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q4: 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),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3177_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,I3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,I3) ) ) ) ) ).

% split_zdiv
tff(fact_3178_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)),Y) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),one_one(word(A))) ) ) ) ) ).

% word_div_sub
tff(fact_3179_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),cons(A,Xa),Xs),Ys) = aa(list(B),list(product_prod(A,B)),case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_jd(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Xa),Xs)),Ys) ).

% zip_Cons1
tff(fact_3180_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),cons(B,Y),Ys)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_je(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_3181_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = foldr(list(A),nat,aTP_Lamp_jf(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_3182_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,W)),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Z),W)))) ) ) ) ).

% norm_power_diff
tff(fact_3183_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_3184_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_3185_transpose_Opelims,axiom,
    ! [A: $tType,Xa: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xa) = Y )
     => ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),Xa)
       => ( ( ( Xa = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( Xa = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) ) )
           => ~ ! [X: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( Xa = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_iw(A,fun(list(A),list(A)))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_iv(A,fun(list(A),list(list(A))))),Xss2))))) )
                   => ~ aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_3186_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Xss: list(list(A))] :
      ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xa),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xa),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xa),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_iw(A,fun(list(A),list(A)))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_iv(A,fun(list(A),list(list(A))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_3187_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = uminus_uminus(int,A2) )
      <=> ( B2 = uminus_uminus(int,one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_3188_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_3189_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_3190_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_3191_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_3192_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_3193_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_3194_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% div_mult_mult1_if
tff(fact_3195_div__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_3196_div__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_3197_div__eq__dividend__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = M )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_3198_div__mult__self__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)),Nb) = M ) ) ).

% div_mult_self_is_m
tff(fact_3199_div__mult__self1__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),M)),Nb) = M ) ) ).

% div_mult_self1_is_m
tff(fact_3200_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_3201_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_3202_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = A2 )
      <=> ( B2 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_3203_div__le__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),M) ).

% div_le_dividend
tff(fact_3204_div__le__mono,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),K)) ) ).

% div_le_mono
tff(fact_3205_div__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),Q4) ).

% div_mult2_eq
tff(fact_3206_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3207_Suc__div__le__mono,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),Nb)) ).

% Suc_div_le_mono
tff(fact_3208_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_3209_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))),M) ).

% times_div_less_eq_dividend
tff(fact_3210_div__times__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),Nb)),M) ).

% div_times_less_eq_dividend
tff(fact_3211_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)),B2)),A2) ).

% div_mult_le
tff(fact_3212_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% div_mult2_eq'
tff(fact_3213_div__le__mono2,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M)) ) ) ).

% div_le_mono2
tff(fact_3214_div__greater__zero__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_3215_div__less__dividend,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),M) ) ) ).

% div_less_dividend
tff(fact_3216_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2)),B2) ) ) ).

% td_gal_lt
tff(fact_3217_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q4)),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ) ) ).

% div_less_iff_less_mult
tff(fact_3218_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),A2) ) ) ).

% zdiv_le_dividend
tff(fact_3219_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_3220_zdiv__mult__self,axiom,
    ! [M: int,A2: int,Nb: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb))),M) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),M)),Nb) ) ) ).

% zdiv_mult_self
tff(fact_3221_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_3222_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_3223_div__if,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb)) ) ).

% div_if
tff(fact_3224_div__nat__eqI,axiom,
    ! [Nb: nat,Q4: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q4)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_3225_split__div,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I3)),J3) )
               => aa(nat,$o,P,I3) ) ) ) ) ) ).

% split_div
tff(fact_3226_dividend__less__div__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_3227_dividend__less__times__div,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)))) ) ).

% dividend_less_times_div
tff(fact_3228_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)),A2)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2)) ) ) ).

% td_gal
tff(fact_3229_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Q4))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3230_divide__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jg(nat,fun(nat,fun(nat,$o)),M),Nb)))) ).

% divide_nat_def
tff(fact_3231_le__div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_3232_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q6)),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q6)))
            & aa(nat,$o,P,Q6) ) ) ) ).

% split_div'
tff(fact_3233_power__sub,axiom,
    ! [Nb: nat,M: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),aa(nat,nat,minus_minus(nat,M),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),Nb)) ) ) ) ).

% power_sub
tff(fact_3234_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,M),Nb)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,Nb),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_3235_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = uminus_uminus(int,one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_3236_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_3237_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_3238_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) != zero_zero(int) )
     => ( sgn_sgn(int,aa(int,int,aa(int,fun(int,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_3239_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M),
            zero_zero(A),
            $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),M)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Nb)))),aa(A,A,minus_minus(A,one_one(A)),Xa))) ) ) ).

% sum_gp
tff(fact_3240_list__encode_Oelims,axiom,
    ! [Xa: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xa) = Y )
     => ( ( ( Xa = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X: nat,Xs2: list(nat)] :
              ( ( Xa = aa(list(nat),list(nat),cons(nat,X),Xs2) )
             => ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_3241_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L4: A,H4: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L4,H4) )
        <=> ( ( ( L = L4 )
              & ( H = H4 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L4),H4) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_3242_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I),set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_3243_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_3244_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_3245_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_3246_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,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_3247_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3248_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Xa,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_3249_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,Xa)) = Xa ) ) ) ).

% cSup_atLeastAtMost
tff(fact_3250_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : set_or1337092689740270186AtMost(A,A2,A2) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_3251_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),insert(A,C2),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3252_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_3253_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_dz(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_3254_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_jh(A,fun(A,A),D)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,A2),D),aa(A,A,minus_minus(A,B2),D)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_3255_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H4: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H4))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),H),H4) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_3256_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D)),set_or1337092689740270186AtMost(A,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_3257_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_ji(A,fun(A,A),D)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_3258_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(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,aa(fun(nat,A),fun(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_3259_list__encode__eq,axiom,
    ! [Xa: list(nat),Y: list(nat)] :
      ( ( aa(list(nat),nat,nat_list_encode,Xa) = aa(list(nat),nat,nat_list_encode,Y) )
    <=> ( Xa = Y ) ) ).

% list_encode_eq
tff(fact_3260_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_3261_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_3262_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3263_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L4: A,H4: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L4,H4)) ) ).

% not_Iic_le_Icc
tff(fact_3264_all__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Nb)
         => aa(nat,$o,P,M5) )
    <=> ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X3) ) ) ).

% all_nat_less
tff(fact_3265_ex__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Nb)
          & aa(nat,$o,P,M5) )
    <=> ? [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X3) ) ) ).

% ex_nat_less
tff(fact_3266_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_3267_atMost__atLeast0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_atMost(nat),Nb) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb) ).

% atMost_atLeast0
tff(fact_3268_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,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3269_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,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hj(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_3270_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3271_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_3272_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_3273_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( set_or1337092689740270186AtMost(nat,M,Nb) = aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3274_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3275_atLeastAtMost__insertL,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) = set_or1337092689740270186AtMost(nat,M,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_3276_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Nb,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or1337092689740270186AtMost(nat,Nb,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_3277_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N7),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite2(nat),N7) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3278_UN__le__add__shift,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_jk(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift
tff(fact_3279_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3280_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3281_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_3282_list__encode_Osimps_I1_J,axiom,
    aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_3283_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3284_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_3285_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(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_3286_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3287_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3288_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_3289_image__Suc__atMost,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,Nb)) ).

% image_Suc_atMost
tff(fact_3290_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3291_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ho(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F,aa(nat,nat,suc,Nb))),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_3292_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_3293_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jl(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% sum.nested_swap
tff(fact_3294_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_jo(fun(nat,A),fun(nat,fun(A,A)),F),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3295_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_3296_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(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_3297_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(aa(nat,$o,aa(nat,fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.head_if
tff(fact_3298_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)))) ) ).

% sum_up_index_split
tff(fact_3299_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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jj(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_3300_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xa: A,Y: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,Xa,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),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),Xa)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_3301_UN__le__eq__Un0,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)))),aa(nat,set(A),M6,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_3302_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xa: A,Y: A] :
          aa(set(A),set(A),image(A,A,aTP_Lamp_jp(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,Xa,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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),Xa),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_3303_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_jq(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_3304_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_jr(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_3305_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_js(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_3306_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_jt(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_3307_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ju(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb),aa(A,A,minus_minus(A,aa(nat,A,F,M)),aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_3308_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jv(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F,Nb)),aa(nat,A,F,M)) ) ) ) ).

% sum_telescope''
tff(fact_3309_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Nb: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),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),Xa),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,Nb),M)))) ) ) ) ).

% sum_power_shift
tff(fact_3310_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),aa(nat,set(nat),set_ord_atMost(nat),Nb)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_3311_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Nb: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Nb))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Nb))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3312_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: A] :
          ( ! [X3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X3: nat] :
                ( aa(set(nat),$o,member(nat,X3),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C2,X3) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3313_list__encode_Osimps_I2_J,axiom,
    ! [Xa: nat,Xs: list(nat)] : aa(list(nat),nat,nat_list_encode,aa(list(nat),list(nat),cons(nat,Xa),Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(list(nat),nat,nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_3314_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) = $ite(Xa = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Nb))))),aa(A,A,minus_minus(A,one_one(A)),Xa))) ) ).

% sum_gp_offset
tff(fact_3315_list__encode_Opelims,axiom,
    ! [Xa: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xa) = Y )
     => ( aa(list(nat),$o,accp(list(nat),nat_list_encode_rel),Xa)
       => ( ( ( Xa = nil(nat) )
           => ( ( Y = zero_zero(nat) )
             => ~ aa(list(nat),$o,accp(list(nat),nat_list_encode_rel),nil(nat)) ) )
         => ~ ! [X: nat,Xs2: list(nat)] :
                ( ( Xa = aa(list(nat),list(nat),cons(nat,X),Xs2) )
               => ( ( Y = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),aa(list(nat),nat,nat_list_encode,Xs2)))) )
                 => ~ aa(list(nat),$o,accp(list(nat),nat_list_encode_rel),aa(list(nat),list(nat),cons(nat,X),Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_3316_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xa: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(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_jx(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xa),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff
tff(fact_3317_product__nth,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),Nb) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_3318_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xa: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(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_jz(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xa),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3319_arctan__add,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(Xa)),arctan(Y)) = arctan(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)))) ) ) ) ).

% arctan_add
tff(fact_3320_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_3321_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_3322_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_3323_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_3324_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( aa(set(A),$o,member(A,I),aa(A,set(A),set_ord_lessThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),K) ) ) ).

% lessThan_iff
tff(fact_3325_nat__mod__eq_H,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),Nb)
     => ( modulo_modulo(nat,A2,Nb) = A2 ) ) ).

% nat_mod_eq'
tff(fact_3326_mod__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
     => ( modulo_modulo(nat,M,Nb) = M ) ) ).

% mod_less
tff(fact_3327_arctan__zero__zero,axiom,
    arctan(zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_3328_arctan__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( ( arctan(Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_3329_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_3330_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_3331_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_3332_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_3333_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_3334_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_3335_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_3336_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_3337_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_3338_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_3339_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),Xa)),aa(A,set(A),set_ord_lessThan(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% lessThan_subset_iff
tff(fact_3340_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_3341_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_3342_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),arctan(Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% zero_less_arctan_iff
tff(fact_3343_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arctan(Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% zero_le_arctan_iff
tff(fact_3344_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_3345_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3346_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : modulo_modulo(A,A2,uminus_uminus(A,one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_3347_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_3348_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_3349_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_3350_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_3351_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% sum.lessThan_Suc
tff(fact_3352_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),insert(A,K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_3353_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K))) = append(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,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_3354_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] : aa(A,set(A),set_ord_lessThan(A),Xa) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_3355_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_3356_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_3357_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_3358_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A6,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),A6),B6),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_3359_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_3360_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_3361_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_3362_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_3363_nat__mod__eq,axiom,
    ! [B2: nat,Nb: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3364_mod__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,Nb)),M) ).

% mod_less_eq_dividend
tff(fact_3365_arctan__monotone_H,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),arctan(Y)) ) ).

% arctan_monotone'
tff(fact_3366_arctan__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),arctan(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% arctan_le_iff
tff(fact_3367_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ka(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_3368_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_kb(fun(B,A),fun(A,fun(B,A)),F),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3),A2) ) ).

% mod_sum_eq
tff(fact_3369_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),modulo_modulo(A,A2,B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_3370_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A2,B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_3371_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_3372_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_3373_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [Nb: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),Nb) = bot_bot(set(A)) )
        <=> ( Nb = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_3374_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_3375_mod__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat,P3: nat,M: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),P3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),P3)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$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_3376_nat__mod__lem,axiom,
    ! [Nb: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),Nb)
      <=> ( modulo_modulo(nat,B2,Nb) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_3377_mod__less__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),modulo_modulo(nat,M,Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_3378_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,aa(nat,fun(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_3379_mod__Suc__le__divisor,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_3380_lessThan__atLeast0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Nb) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ).

% lessThan_atLeast0
tff(fact_3381_range__subset__card,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A),D: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),A2,B2)),set_or1337092689740270186AtMost(word(A),C2,D))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),D)
              & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),B2),A2)),aa(word(A),word(A),minus_minus(word(A),D),C2)) ) ) ) ) ).

% range_subset_card
tff(fact_3382_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,$o,aa(nat,fun(nat,$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_3383_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M)),aa(A,set(A),set_ord_lessThan(A),Nb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Nb) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3384_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_3385_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_atMost(nat),K) ).

% lessThan_Suc_atMost
tff(fact_3386_mod__nat__sub,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( modulo_modulo(nat,aa(nat,nat,minus_minus(nat,Xa),Y),Z) = aa(nat,nat,minus_minus(nat,Xa),Y) ) ) ).

% mod_nat_sub
tff(fact_3387_mod__if,axiom,
    ! [M: nat,Nb: nat] :
      modulo_modulo(nat,M,Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb),M,modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ).

% mod_if
tff(fact_3388_mod__geq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$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_3389_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( modulo_modulo(nat,M,D) = zero_zero(nat) )
     => ? [Q3: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D),Q3) ) ).

% mod_eq_0D
tff(fact_3390_le__mod__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3391_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),Nb) = bot_bot(set(nat)) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3392_nat__mod__eq__iff,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xa,Nb) = modulo_modulo(nat,Y,Nb) )
    <=> ? [Q12: nat,Q23: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q12)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q23)) ) ).

% nat_mod_eq_iff
tff(fact_3393_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_3394_lessThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% lessThan_Suc
tff(fact_3395_finite__nat__iff__bounded,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S3)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S3),aa(nat,set(nat),set_ord_lessThan(nat),K3)) ) ).

% finite_nat_iff_bounded
tff(fact_3396_finite__nat__bounded,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S3)
     => ? [K2: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S3),aa(nat,set(nat),set_ord_lessThan(nat),K2)) ) ).

% finite_nat_bounded
tff(fact_3397_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3398_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A2,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3399_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_mult1_eq
tff(fact_3400_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),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_3401_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),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_3402_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) ) ).

% mod_div_decomp
tff(fact_3403_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_3404_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_3405_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_3406_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),modulo_modulo(A,A2,B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_3407_zmde,axiom,
    ! [A: $tType] :
      ( ( group_add(A)
        & semiring_modulo(A) )
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) ) ).

% zmde
tff(fact_3408_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = modulo_modulo(A,A2,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_3409_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_3410_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_3411_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = modulo_modulo(A,A2,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_3412_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)),set_or1337092689740270186AtMost(int,M,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3413_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),insert(int,I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_3414_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A2)),aa(A,set(A),set_ord_lessThan(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3415_mod__le__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_3416_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_3417_div__less__mono,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_3418_mod__nat__add,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z)
       => ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y),Z) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_3419_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xa,Nb) = modulo_modulo(nat,Y,Nb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
       => ? [Q3: nat] : Xa = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3420_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,Nb: nat] :
      ( ( modulo_modulo(nat,M,Q4) = modulo_modulo(nat,Nb,Q4) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ~ ! [S: nat] : Nb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),S)) ) ) ).

% mod_eq_nat2E
tff(fact_3421_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,Nb: nat] :
      ( ( modulo_modulo(nat,M,Q4) = modulo_modulo(nat,Nb,Q4) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
       => ~ ! [S: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),S)) ) ) ).

% mod_eq_nat1E
tff(fact_3422_mod__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) = 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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb),Q4))),modulo_modulo(nat,M,Nb)) ).

% mod_mult2_eq
tff(fact_3423_div__mod__decomp,axiom,
    ! [A3: nat,Nb: nat] : A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Nb)),Nb)),modulo_modulo(nat,A3,Nb)) ).

% div_mod_decomp
tff(fact_3424_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_3425_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_kc(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.nat_diff_reindex
tff(fact_3426_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_jk(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift_strict
tff(fact_3427_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Nb: A] :
          ( ! [X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X)),aa(A,nat,P,X))
         => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),aa(A,set(A),set_ord_lessThan(A),Nb))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),aa(A,set(A),set_ord_lessThan(A),Nb))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_kd(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_3428_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_3429_mod__lemma,axiom,
    ! [C2: nat,R2: nat,B2: nat,Q4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),B2)
       => aa(nat,$o,aa(nat,fun(nat,$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,Q4,C2))),R2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)) ) ) ).

% mod_lemma
tff(fact_3430_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,$o,P,M) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I3)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_3431_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_3432_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_3433_image__Suc__lessThan,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),Nb) ).

% image_Suc_lessThan
tff(fact_3434_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3435_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ho(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F,M)),aa(nat,A,F,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3436_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ij(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3437_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F: fun(nat,A),Nb: nat,R2: A] : aa(A,A,minus_minus(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ke(fun(nat,A),fun(A,fun(nat,A)),F),R2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sumr_diff_mult_const2
tff(fact_3438_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_3439_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3440_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_kf(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% sum.nat_group
tff(fact_3441_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_kg(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.nested_swap'
tff(fact_3442_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_3443_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( 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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3444_aset_I7_J,axiom,
    ! [D4: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)) ) ) ) ).

% aset(7)
tff(fact_3445_aset_I5_J,axiom,
    ! [D4: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,Ta),A3)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_3446_aset_I4_J,axiom,
    ! [D4: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,Ta),A3)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa4) ) ) )
           => ( ( X4 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_3447_aset_I3_J,axiom,
    ! [D4: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A3)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa4) ) ) )
           => ( ( X4 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_3448_bset_I7_J,axiom,
    ! [D4: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,Ta),B4)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,minus_minus(int,X4),D4)) ) ) ) ) ).

% bset(7)
tff(fact_3449_bset_I5_J,axiom,
    ! [D4: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,X4),D4)),Ta) ) ) ) ).

% bset(5)
tff(fact_3450_bset_I4_J,axiom,
    ! [D4: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,Ta),B4)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( ( X4 != Ta )
             => ( aa(int,int,minus_minus(int,X4),D4) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_3451_bset_I3_J,axiom,
    ! [D4: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,minus_minus(int,Ta),one_one(int))),B4)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( ( X4 = Ta )
             => ( aa(int,int,minus_minus(int,X4),D4) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_3452_periodic__finite__ex,axiom,
    ! [D: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P,X)
          <=> aa(int,$o,P,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
       => ( ? [X_12: int] : aa(int,$o,P,X_12)
        <=> ? [X3: int] :
              ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D))
              & aa(int,$o,P,X3) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3453_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),R2: word(A),Y: word(A),S2: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),Xa,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),R2)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S2)),one_one(word(A)))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),R2)),one_one(word(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S2)),one_one(word(A))))
             => ( ( S2 != zero_zero(word(A)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),R2),S2) ) ) ) ) ) ).

% word_subset_less
tff(fact_3454_Suc__times__mod__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3455_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% one_diff_power_eq
tff(fact_3456_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xa),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% power_diff_1_eq
tff(fact_3457_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Nb: nat] :
          ( ( Xa != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb)),one_one(A))),aa(A,A,minus_minus(A,Xa),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3458_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.atMost_shift
tff(fact_3459_word__atLeastLessThan__Suc__atLeastAtMost__union,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A),L: word(A),U: word(A)] :
          ( ( M != uminus_uminus(word(A),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),L),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),M),U)
             => ( aa(set(word(A)),set(word(A)),aa(set(word(A)),fun(set(word(A)),set(word(A))),sup_sup(set(word(A))),set_or1337092689740270186AtMost(word(A),L,M)),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)) = set_or1337092689740270186AtMost(word(A),L,U) ) ) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost_union
tff(fact_3460_aset_I8_J,axiom,
    ! [D4: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A3)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)) ) ) ) ).

% aset(8)
tff(fact_3461_aset_I6_J,axiom,
    ! [D4: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A3)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_3462_bset_I8_J,axiom,
    ! [D4: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( aa(set(int),$o,member(int,aa(int,int,minus_minus(int,Ta),one_one(int))),B4)
       => ! [X4: int] :
            ( ! [Xa4: int] :
                ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,minus_minus(int,X4),D4)) ) ) ) ) ).

% bset(8)
tff(fact_3463_bset_I6_J,axiom,
    ! [D4: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X4: int] :
          ( ! [Xa4: int] :
              ( aa(set(int),$o,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,X4),D4)),Ta) ) ) ) ).

% bset(6)
tff(fact_3464_cppi,axiom,
    ! [D4: int,P: fun(int,$o),P2: fun(int,$o),A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( aa(set(int),$o,member(int,Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb3: int] :
                      ( aa(set(int),$o,member(int,Xb3),A3)
                     => ( X != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) )
         => ( ! [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),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P2,X3) )
                | ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa3: int] :
                        ( aa(set(int),$o,member(int,Xa3),A3)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa3),X3)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3465_cpmi,axiom,
    ! [D4: int,P: fun(int,$o),P2: fun(int,$o),B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( aa(set(int),$o,member(int,Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb3: int] :
                      ( aa(set(int),$o,member(int,Xb3),B4)
                     => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X),D4)) ) )
         => ( ! [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),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P2,X3) )
                | ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa3: int] :
                        ( aa(set(int),$o,member(int,Xa3),B4)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X3)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3466_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Nb)),
                $ite(modulo_modulo(nat,B4,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_3467_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kh(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ki(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),aa(nat,set(nat),set_ord_lessThan(nat),M)) ) ).

% lemma_termdiff1
tff(fact_3468_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb))),aa(A,A,minus_minus(A,one_one(A)),Xa))) ) ).

% sum_gp_strict
tff(fact_3469_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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,Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_kj(A,fun(nat,fun(A,fun(nat,A))),Xa),Nb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% power_diff_sumr2
tff(fact_3470_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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,Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_kk(A,fun(nat,fun(A,fun(nat,A))),Xa),Nb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb)))) ) ).

% diff_power_eq_sum
tff(fact_3471_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,Nb: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z4),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),B3),Z4)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ).

% polyfun_linear_factor_root
tff(fact_3472_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Nb: nat,A2: A] :
        ? [B3: fun(nat,A)] :
        ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z4),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),B3),Z4)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% polyfun_linear_factor
tff(fact_3473_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),aa(nat,set(nat),set_ord_lessThan(nat),Nb)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3474_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F: fun(nat,A),K4: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,P4)),K4) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),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)),K4)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3475_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Nb: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_kl(A,fun(nat,fun(nat,A)),Xa),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% one_diff_power_eq'
tff(fact_3476_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hi(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3477_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_km(nat,real)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3478_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = $ite(Xs = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_3479_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,uminus_uminus(real,Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_3480_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(nat,real,semiring_1_of_nat(real),Nb))),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_3481_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_3482_length__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rev
tff(fact_3483_exp__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),exp(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% exp_le_cancel_iff
tff(fact_3484_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( exp(A,zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_3485_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_3486_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_3487_exp__eq__one__iff,axiom,
    ! [Xa: real] :
      ( ( exp(real,Xa) = one_one(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_3488_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3489_sorted__wrt__rev__linord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,aTP_Lamp_gd(A,fun(A,$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_3490_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% one_less_exp_iff
tff(fact_3491_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_3492_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_3493_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),exp(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% one_le_exp_iff
tff(fact_3494_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( exp(real,ln_ln(real,Xa)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% exp_ln_iff
tff(fact_3495_exp__ln,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( exp(real,ln_ln(real,Xa)) = Xa ) ) ).

% exp_ln
tff(fact_3496_Word_Omod__minus__cong,axiom,
    ! [B2: int,B6: int,Xa: int,X6: int,Y: int,Y7: int,Z6: int] :
      ( ( B2 = B6 )
     => ( ( modulo_modulo(int,Xa,B6) = modulo_modulo(int,X6,B6) )
       => ( ( modulo_modulo(int,Y,B6) = modulo_modulo(int,Y7,B6) )
         => ( ( aa(int,int,minus_minus(int,X6),Y7) = Z6 )
           => ( modulo_modulo(int,aa(int,int,minus_minus(int,Xa),Y),B2) = modulo_modulo(int,Z6,B6) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_3497_mod__plus__cong,axiom,
    ! [B2: int,B6: int,Xa: int,X6: int,Y: int,Y7: int,Z6: int] :
      ( ( B2 = B6 )
     => ( ( modulo_modulo(int,Xa,B6) = modulo_modulo(int,X6,B6) )
       => ( ( modulo_modulo(int,Y,B6) = modulo_modulo(int,Y7,B6) )
         => ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X6),Y7) = Z6 )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y),B2) = modulo_modulo(int,Z6,B6) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_3498_mod__word__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Va2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),Va2)
         => ( modulo_modulo(word(A),W,Va2) = W ) ) ) ).

% mod_word_less
tff(fact_3499_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_3500_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Xa))),exp(real,real_V7770717601297561774m_norm(A,Xa))) ) ).

% norm_exp
tff(fact_3501_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,A3)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),exp(A,A3)) ) ).

% exp_times_arg_commute
tff(fact_3502_zip__rev,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( zip(A,B,aa(list(A),list(A),rev(A),Xs),aa(list(B),list(B),rev(B),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),zip(A,B,Xs,Ys)) ) ) ).

% zip_rev
tff(fact_3503_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : exp(A,Xa) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_3504_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),rev(A),Xs))
    <=> sorted_wrt(A,aTP_Lamp_kn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_3505_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_3506_exp__gt__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),exp(real,Xa)) ).

% exp_gt_zero
tff(fact_3507_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X: real] : exp(real,X) = Y ) ).

% exp_total
tff(fact_3508_exp__ge__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),exp(real,Xa)) ).

% exp_ge_zero
tff(fact_3509_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_3510_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xa)),exp(A,Y)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) ) ).

% mult_exp_exp
tff(fact_3511_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) )
         => ( exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xa)),exp(A,Y)) ) ) ) ).

% exp_add_commuting
tff(fact_3512_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,M,K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_3513_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),modulo_modulo(int,K,L)) ) ).

% neg_mod_bound
tff(fact_3514_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,K,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_3515_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,K,uminus_uminus(int,L)) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_3516_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,uminus_uminus(int,K),L) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_3517_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo(int,M,D) = zero_zero(int) )
     => ? [Q3: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q3) ) ).

% zmod_eq_0D
tff(fact_3518_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo(int,M,D) = zero_zero(int) )
    <=> ? [Q6: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q6) ) ).

% zmod_eq_0_iff
tff(fact_3519_word__mod__less__divisor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Nb)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),modulo_modulo(word(A),M,Nb)),Nb) ) ) ).

% word_mod_less_divisor
tff(fact_3520_udvd__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] :
          ( udvd(A,Va2,W)
         => ( modulo_modulo(word(A),W,Va2) = zero_zero(word(A)) ) ) ) ).

% udvd_imp_mod_eq_0
tff(fact_3521_mod__eq__0__imp__udvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Va2: word(A)] :
          ( ( modulo_modulo(word(A),W,Va2) = zero_zero(word(A)) )
         => udvd(A,Va2,W) ) ) ).

% mod_eq_0_imp_udvd
tff(fact_3522_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),A2: A] : foldr(B,A,F,Xs,A2) = foldl(A,B,aTP_Lamp_ko(fun(B,fun(A,A)),fun(A,fun(B,A)),F),A2,aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_foldl
tff(fact_3523_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,A)),A2: A,Xs: list(B)] : foldl(A,B,F,A2,Xs) = foldr(B,A,aTP_Lamp_kp(fun(A,fun(B,A)),fun(B,fun(A,A)),F),aa(list(B),list(B),rev(B),Xs),A2) ).

% foldl_conv_foldr
tff(fact_3524_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xa)) ) ).

% exp_gt_one
tff(fact_3525_exp__ge__add__one__self,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),exp(real,Xa)) ).

% exp_ge_add_one_self
tff(fact_3526_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xa)),exp(A,uminus_uminus(A,Xa))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_3527_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Nb: nat] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xa)),Nb) ) ).

% exp_of_nat2_mult
tff(fact_3528_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xa: A] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),Xa)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xa)),Nb) ) ).

% exp_of_nat_mult
tff(fact_3529_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,A2,B2)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),modulo_modulo(int,A2,B2)) ) ) ).

% neg_mod_conj
tff(fact_3530_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A2,B2))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,A2,B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_3531_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_3532_int__mod__ge,axiom,
    ! [A2: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),Nb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),modulo_modulo(int,A2,Nb)) ) ) ).

% int_mod_ge
tff(fact_3533_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3534_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_3535_int__mod__lem,axiom,
    ! [Nb: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),Nb) )
      <=> ( modulo_modulo(int,B2,Nb) = B2 ) ) ) ).

% int_mod_lem
tff(fact_3536_int__mod__eq,axiom,
    ! [B2: int,Nb: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$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_3537_int__mod__le_H,axiom,
    ! [B2: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,minus_minus(int,B2),Nb))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,B2,Nb)),aa(int,int,minus_minus(int,B2),Nb)) ) ).

% int_mod_le'
tff(fact_3538_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(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_3539_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(int,A2,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_3540_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A2,B2))
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% nonneg_mod_div
tff(fact_3541_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B4)
     => ( aa(int,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Nb)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_3542_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_3543_word__mod__div__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Nb),B2)),B2)),modulo_modulo(word(A),Nb,B2)) = Nb ) ).

% word_mod_div_equality
tff(fact_3544_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),exp(real,Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_3545_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,minus_minus(real,Y),one_one(real)))
          & ( exp(real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3546_ln__ge__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),ln_ln(real,Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Y)),Xa) ) ) ).

% ln_ge_iff
tff(fact_3547_drop__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ).

% drop_rev
tff(fact_3548_rev__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),drop(A,I,Xs)) = take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),I),aa(list(A),list(A),rev(A),Xs)) ).

% rev_drop
tff(fact_3549_rev__take,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),take(A,I,Xs)) = drop(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),I),aa(list(A),list(A),rev(A),Xs)) ).

% rev_take
tff(fact_3550_take__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),drop(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ).

% take_rev
tff(fact_3551_ln__x__over__x__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),ln_ln(real,Y)),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),ln_ln(real,Xa)),Xa)) ) ) ).

% ln_x_over_x_mono
tff(fact_3552_int__mod__ge_H,axiom,
    ! [B2: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),Nb)),modulo_modulo(int,B2,Nb)) ) ) ).

% int_mod_ge'
tff(fact_3553_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3554_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$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_3555_rev__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,Nb))) ) ) ).

% rev_nth
tff(fact_3556_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),rev(A),list_update(A,Xs,K,Y)) = list_update(A,aa(list(A),list(A),rev(A),Xs),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_3557_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = exp(A,Xa) ) ) ) ).

% exp_divide_power_eq
tff(fact_3558_mod__power__lem,axiom,
    ! [A2: int,Nb: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$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,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb),zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Nb)) ) ) ).

% mod_power_lem
tff(fact_3559_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q4: 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),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3560_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q4: 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),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3561_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,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_3562_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( modulo_modulo(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_3563_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( modulo_modulo(int,uminus_uminus(int,one_one(int)),B2) = aa(int,int,minus_minus(int,B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_3564_mod__add__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_3565_mod__sub__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( modulo_modulo(int,aa(int,int,minus_minus(int,Xa),Y),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Xa),aa(int,int,minus_minus(int,Xa),Y),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_3566_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( 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,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2),C2))),modulo_modulo(int,A2,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_3567_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),uminus_uminus(int,A2)),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),uminus_uminus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,minus_minus(int,uminus_uminus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_3568_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),uminus_uminus(int,B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),uminus_uminus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,minus_minus(int,uminus_uminus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_3569_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))),aa(nat,A,nth(A,Xs),I3)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_3570_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_3571_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I3)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_3572_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),map(list(A),nat,size_size(list(A)),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_3573_finite__int__iff__bounded,axiom,
    ! [S3: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S3)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S3)),aa(int,set(int),set_ord_lessThan(int),K3)) ) ).

% finite_int_iff_bounded
tff(fact_3574_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Nb)),
                $ite(modulo_modulo(int,B4,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_3575_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K)),modulo_modulo(int,Nb,K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_3576_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K)),modulo_modulo(int,Nb,K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_3577_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( foldr(A,A,ord_max(A),Xs,Y) = $ite(Xs = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_3578_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_kq(nat,fun(list(A),$o),I)),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_3579_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( map(list(A),A,aTP_Lamp_kr(nat,fun(list(A),A),I),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_kq(nat,fun(list(A),$o),I)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_3580_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Nb: nat] :
      ( ( Xa != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T6))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xa))
            & ( exp(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ks(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T6)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3581_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_kq(nat,fun(list(A),$o),I)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_3582_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Nb = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = Nb ) )
       => ( transpose(A,Xs) = map(nat,list(A),aTP_Lamp_ku(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_3583_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,$o),aTP_Lamp_kv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),Xs) ).

% filter_filter
tff(fact_3584_of__nat__fact,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),semiring_char_0_fact(nat,Nb)) = semiring_char_0_fact(A,Nb) ) ).

% of_nat_fact
tff(fact_3585_length__concat__rev,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) = aa(list(A),nat,size_size(list(A)),concat(A,Xs)) ).

% length_concat_rev
tff(fact_3586_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)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_kw(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ).

% set_filter
tff(fact_3587_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3588_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3589_take__upt,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)),Nb)
     => ( take(nat,M,upt(I,Nb)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)) ) ) ).

% take_upt
tff(fact_3590_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_3591_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_3592_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_3593_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K) )
     => ( append(nat,upt(I,J),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_3594_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_3595_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_3596_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( last(nat,upt(I,J)) = aa(nat,nat,minus_minus(nat,J),one_one(nat)) ) ) ).

% last_upt
tff(fact_3597_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_3598_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3599_sum__list__upt,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( groups8242544230860333062m_list(nat,upt(M,Nb)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_kx(nat,nat)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ) ).

% sum_list_upt
tff(fact_3600_map__Suc__upt,axiom,
    ! [M: nat,Nb: nat] : map(nat,nat,suc,upt(M,Nb)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Nb)) ).

% map_Suc_upt
tff(fact_3601_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_3602_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : map(nat,nat,aTP_Lamp_ky(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_3603_removeAll__filter__not__eq,axiom,
    ! [A: $tType,Xa: A] : removeAll(A,Xa) = filter2(A,aTP_Lamp_kz(A,fun(A,$o),Xa)) ).

% removeAll_filter_not_eq
tff(fact_3604_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_3605_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_3606_sorted__filter_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,P),L)) ) ) ).

% sorted_filter'
tff(fact_3607_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_zero
tff(fact_3608_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_gt_zero
tff(fact_3609_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Nb)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_3610_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_1
tff(fact_3611_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Nb)) ) ) ).

% fact_mono
tff(fact_3612_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Nb: nat,Ns: list(nat),Q4: nat] :
      ( ( aa(list(nat),list(nat),cons(nat,M),aa(list(nat),list(nat),cons(nat,Nb),Ns)) = upt(M,Q4) )
    <=> ( aa(list(nat),list(nat),cons(nat,Nb),Ns) = upt(aa(nat,nat,suc,M),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_3613_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_3614_filter__conv__foldr,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = foldr(A,list(A),aTP_Lamp_la(fun(A,$o),fun(A,fun(list(A),list(A))),P),Xs,nil(A)) ).

% filter_conv_foldr
tff(fact_3615_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_eh(fun(A,$o),fun(A,$o),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_3616_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_lb(fun(list(A),A),fun(list(A),fun(A,$o)),G),Xs)),Xs)) ) ).

% sorted_same
tff(fact_3617_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_3618_map__add__upt,axiom,
    ! [Nb: nat,M: nat] : map(nat,nat,aTP_Lamp_ky(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_3619_sorted__wrt__upt,axiom,
    ! [M: nat,Nb: nat] : sorted_wrt(nat,ord_less(nat),upt(M,Nb)) ).

% sorted_wrt_upt
tff(fact_3620_sorted__upt,axiom,
    ! [M: nat,Nb: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,Nb)) ).

% sorted_upt
tff(fact_3621_replicate__length__filter,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Xa)),Xs)),Xa) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Xa)),Xs) ).

% replicate_length_filter
tff(fact_3622_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( upt(I,J) = append(nat,Xs,Ys) )
      <=> ? [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),J)
            & ( upt(I,K3) = Xs )
            & ( upt(K3,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_3623_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_lc(list(A),$o)),Xs)) = concat(A,Xs) ).

% concat_filter_neq_Nil
tff(fact_3624_length__filter__less,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xa)
       => aa(nat,$o,aa(nat,fun(nat,$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_3625_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B),P: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F,aa(list(B),list(B),filter2(B,P),Xs))) ) ) ).

% sorted_filter
tff(fact_3626_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3627_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Nb)) ) ) ) ).

% fact_less_mono
tff(fact_3628_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( modulo_modulo(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3629_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Nb)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),Nb))) ) ).

% fact_le_power
tff(fact_3630_enumerate__eq__zip,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : enumerate(A,Nb,Xs) = zip(nat,A,upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_3631_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F,aa(list(B),list(B),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_ld(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F),G),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_3632_map__decr__upt,axiom,
    ! [M: nat,Nb: nat] : map(nat,nat,aTP_Lamp_le(nat,nat),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = upt(M,Nb) ).

% map_decr_upt
tff(fact_3633_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F: fun(B,A),P: fun(B,$o),Xs: list(B)] : groups8242544230860333062m_list(A,map(B,A,F,aa(list(B),list(B),filter2(B,P),Xs))) = groups8242544230860333062m_list(A,map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_lf(fun(B,A),fun(fun(B,$o),fun(B,A)),F),P),Xs)) ) ).

% sum_list_map_filter'
tff(fact_3634_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( append(nat,upt(zero_zero(nat),I),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_3635_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_3636_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = 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_3637_atLeast__upt,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Nb)) ).

% atLeast_upt
tff(fact_3638_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F: fun(A,nat),P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),groups8242544230860333062m_list(nat,map(A,nat,F,aa(list(A),list(A),filter2(A,P),Xs)))),groups8242544230860333062m_list(nat,map(A,nat,F,Xs))) ).

% sum_list_filter_le_nat
tff(fact_3639_map__replicate__trivial,axiom,
    ! [A: $tType,Xa: A,I: nat] : map(nat,A,aTP_Lamp_lg(A,fun(nat,A),Xa),upt(zero_zero(nat),I)) = replicate(A,I,Xa) ).

% map_replicate_trivial
tff(fact_3640_enumerate__map__upt,axiom,
    ! [A: $tType,Nb: nat,F: fun(nat,A),M: nat] : enumerate(A,Nb,map(nat,A,F,upt(Nb,M))) = map(nat,product_prod(nat,A),aTP_Lamp_lh(fun(nat,A),fun(nat,product_prod(nat,A)),F),upt(Nb,M)) ).

% enumerate_map_upt
tff(fact_3641_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(A),P: fun(A,$o),F: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
             => ( ~ aa(A,$o,P,X)
               => ( aa(A,B,F,X) = zero_zero(B) ) ) )
         => ( groups8242544230860333062m_list(B,map(A,B,F,aa(list(A),list(A),filter2(A,P),Xs))) = groups8242544230860333062m_list(B,map(A,B,F,Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_3642_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B),P: fun(B,$o),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
         => ( aa(B,$o,P,Xa)
           => ( aa(list(B),list(B),filter2(B,P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),Xa),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),Xa),aa(list(B),list(B),filter2(B,P),Xs)) ) ) ) ) ).

% filter_insort
tff(fact_3643_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,Xa),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = Xa )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_3644_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_li(A,fun(A,$o),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_3645_atMost__upto,axiom,
    ! [Nb: nat] : aa(nat,set(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_3646_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,aa(nat,fun(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_3647_map__upt__Suc,axiom,
    ! [A: $tType,F: fun(nat,A),Nb: nat] : map(nat,A,F,upt(zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(list(A),list(A),cons(A,aa(nat,A,F,zero_zero(nat))),map(nat,A,aTP_Lamp_lj(fun(nat,A),fun(nat,A),F),upt(zero_zero(nat),Nb))) ).

% map_upt_Suc
tff(fact_3648_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : map(nat,A,nth(A,Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_3649_enumerate__replicate__eq,axiom,
    ! [A: $tType,Nb: nat,M: nat,A2: A] : enumerate(A,Nb,replicate(A,M,A2)) = map(nat,product_prod(nat,A),aTP_Lamp_lk(A,fun(nat,product_prod(nat,A)),A2),upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M))) ).

% enumerate_replicate_eq
tff(fact_3650_filter__nth__ex__nth,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs)))
     => ? [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,aa(list(A),list(A),filter2(A,P),Xs)),Nb) = aa(nat,A,nth(A,Xs),M2) )
          & ( aa(list(A),list(A),filter2(A,P),take(A,M2,Xs)) = take(A,Nb,aa(list(A),list(A),filter2(A,P),Xs)) ) ) ) ).

% filter_nth_ex_nth
tff(fact_3651_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Nb: nat,M: nat,F: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,Nb),M))
     => ( aa(nat,A,nth(A,map(nat,A,F,upt(M,Nb))),I) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_3652_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
      ( ( upt(L,H) = append(nat,Is1,aa(list(nat),list(nat),cons(nat,I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa(nat,nat,suc,I),H) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),I)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_3653_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_3654_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J),append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_3655_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,M: nat,F: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,minus_minus(nat,Nb),M) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(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,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( map(nat,A,F,upt(M,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_3656_map__nth__upt__drop__take__conv,axiom,
    ! [A: $tType,N7: nat,L: list(A),M6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),aa(list(A),nat,size_size(list(A)),L))
     => ( map(nat,A,nth(A,L),upt(M6,N7)) = drop(A,M6,take(A,N7,L)) ) ) ).

% map_nth_upt_drop_take_conv
tff(fact_3657_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_3658_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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_3659_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Nb: nat] : comm_s3205402744901411588hammer(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),uminus_uminus(A,one_one(A))),Nb)),semiring_char_0_fact(A,Nb)) ) ).

% pochhammer_same
tff(fact_3660_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_ll(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_lm(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_ln(list(B),$o)),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_3661_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = map(list(A),A,aTP_Lamp_kr(nat,fun(list(A),A),I),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_kq(nat,fun(list(A),$o),I)),Xs)) ) ) ).

% nth_transpose
tff(fact_3662_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_jf(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_lc(list(A),$o)),Xs)) ).

% transpose_max_length
tff(fact_3663_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: real,Nb: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xa = zero_zero(real) )
         => ( ( Nb != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_lo(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xa),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3664_Maclaurin__lemma,axiom,
    ! [H: real,F: fun(real,real),J: fun(nat,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ? [B10: real] : aa(real,real,F,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_lp(real,fun(fun(nat,real),fun(nat,real)),H),J)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),B10),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb)),semiring_char_0_fact(real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_3665_Maclaurin__exp__le,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T6: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xa))
      & ( exp(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ks(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T6)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_3666_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,Nb,F),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_lq(nat,fun(fun(nat,A),fun(array(A),assn)),Nb),F)) ) ).

% make_rule
tff(fact_3667_remove__rev__alt__def,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : remove_rev(A,Xa,Xs) = aa(list(A),list(A),filter2(A,aTP_Lamp_li(A,fun(A,$o),Xa)),aa(list(A),list(A),rev(A),Xs)) ).

% remove_rev_alt_def
tff(fact_3668_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] :
          aa(nat,A,gbinomial(A,A2),K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_lr(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_3669_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_iv(A,fun(list(A),list(list(A))))),Xss)) = map(list(A),list(A),tl(A),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_lc(list(A),$o)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_3670_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_3671_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_3672_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_3673_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_3674_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_3675_fact__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Nb)) ) ).

% fact_mono_nat
tff(fact_3676_fact__ge__self,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_self
tff(fact_3677_fact__less__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_3678_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),tl(A),Xs)) ) ) ).

% sorted_tl
tff(fact_3679_take__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),take(A,aa(nat,nat,suc,Nb),Xs)) ).

% take_tl
tff(fact_3680_drop__Suc,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,aa(nat,nat,suc,Nb),Xs) = drop(A,Nb,aa(list(A),list(A),tl(A),Xs)) ).

% drop_Suc
tff(fact_3681_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_ls(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_3682_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),append(A,Xs,Ys)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys),aTP_Lamp_lt(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_3683_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_3684_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_3685_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_3686_tl__subset,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( ( Xs != nil(A) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))),A3) ) ) ).

% tl_subset
tff(fact_3687_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,$o,aa(nat,fun(nat,$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)) )
       => ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),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)),N2)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N2)) ) ) ) ).

% list_take_induct_tl2
tff(fact_3688_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,nat,minus_minus(nat,Nb),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3689_filter__upt__take__conv,axiom,
    ! [A: $tType,P: fun(A,$o),M: nat,L: list(A),Nb: nat] : aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_lu(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),P),M),L)),upt(Nb,M)) = aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_lv(fun(A,$o),fun(list(A),fun(nat,$o)),P),L)),upt(Nb,M)) ).

% filter_upt_take_conv
tff(fact_3690_fact__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3691_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Nb),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_3692_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_3693_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_3694_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_3695_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_3696_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_3697_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A2),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3698_List_Onth__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(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_3699_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),U3)
     => ( ! [I2: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),I2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),U3) )
           => ~ aa(nat,$o,P,I2) )
       => ( 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),U3)) ) ) ) ).

% upt_filter_extend
tff(fact_3700_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_3701_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_3702_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3703_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_lw(A,fun(nat,A),A2)),aa(nat,set(nat),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_3704_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3705_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% gbinomial_factors
tff(fact_3706_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),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_3707_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),uminus_uminus(A,one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(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),uminus_uminus(A,one_one(A))),Nb)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,uminus_uminus(A,aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),Nb)) ) ).

% gbinomial_index_swap
tff(fact_3708_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(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),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_3709_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(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_3710_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),uminus_uminus(A,one_one(A))),K)),comm_s3205402744901411588hammer(A,uminus_uminus(A,A2),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3711_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_lx(A,fun(nat,A),A2)),aa(nat,set(nat),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),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_3712_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ly(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_3713_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,Xa: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(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_lz(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(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_ma(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_3714_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,$o),aTP_Lamp_lv(fun(A,$o),fun(list(A),fun(nat,$o)),P),L)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = append(nat,Js,aa(list(nat),list(nat),cons(nat,J),nil(nat))) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),I)
       => ( aa(nat,$o,aa(nat,fun(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_3715_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3716_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,Xa: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(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_lz(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(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_mb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3717_array__of__list__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : array_of_list(A,Xs) = array_make(A,aa(list(A),nat,size_size(list(A)),Xs),nth(A,Xs)) ) ).

% array_of_list_make
tff(fact_3718_array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xa: A] : array_new(A,Nb,Xa) = array_make(A,Nb,aTP_Lamp_mc(A,fun(nat,A),Xa)) ) ).

% array_make
tff(fact_3719_signed_Ostable__sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Sk: fun(fun(A,word(B)),fun(list(A),list(A)))] :
          ( stable_sort_key(A,word(B),Sk)
        <=> ! [F7: fun(A,word(B)),Xs3: list(A),K3: word(B)] : aa(list(A),list(A),filter2(A,aa(word(B),fun(A,$o),aTP_Lamp_md(fun(A,word(B)),fun(word(B),fun(A,$o)),F7),K3)),aa(list(A),list(A),aa(fun(A,word(B)),fun(list(A),list(A)),Sk,F7),Xs3)) = aa(list(A),list(A),filter2(A,aa(word(B),fun(A,$o),aTP_Lamp_md(fun(A,word(B)),fun(word(B),fun(A,$o)),F7),K3)),Xs3) ) ) ).

% signed.stable_sort_key_def
tff(fact_3720_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_me(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3721_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_3722_binomial__1,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ).

% binomial_1
tff(fact_3723_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_3724_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_3725_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_3726_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_3727_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_3728_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
     => ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_3729_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_3730_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_3731_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,binomial(Nb),aa(nat,nat,minus_minus(nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_3732_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_3733_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$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_3734_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K)) ) ).

% zero_less_binomial
tff(fact_3735_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_3736_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_3737_choose__mult,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_3738_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_3739_sum__choose__upper,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_mf(nat,fun(nat,nat),M)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_3740_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_3741_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Nb),K)))),aa(nat,nat,binomial(Nb),K)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_3742_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_mg(nat,fun(nat,nat),R2)),aa(nat,set(nat),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_3743_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_mh(nat,fun(nat,nat),Nb)),aa(nat,set(nat),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_3744_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_mh(nat,fun(nat,nat),Nb)),aa(nat,set(nat),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_3745_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_3746_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$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_3747_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3748_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3749_sum__choose__diagonal,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_mi(nat,fun(nat,fun(nat,nat)),M),Nb)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),M) ) ) ).

% sum_choose_diagonal
tff(fact_3750_vandermonde,axiom,
    ! [M: nat,Nb: nat,R2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_mj(nat,fun(nat,fun(nat,fun(nat,nat))),M),Nb),R2)),aa(nat,set(nat),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_3751_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_3752_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Nb),K))) ) ) ) ).

% fact_binomial
tff(fact_3753_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Nb),K)))) ) ) ) ).

% binomial_fact
tff(fact_3754_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,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_mk(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ).

% binomial
tff(fact_3755_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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ml(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% binomial_ring
tff(fact_3756_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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_mm(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% pochhammer_binomial_sum
tff(fact_3757_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( ( Nb != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_mn(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3758_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,$o),Xs: list(B)] : map(B,A,F,aa(list(B),list(B),filter2(B,P),Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_mo(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F),P),Xs) ).

% map_filter_map_filter
tff(fact_3759_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),map(list(A),nat,size_size(list(A)),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_lc(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_3760_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),Xa) = Y )
         => ( ( ( Xa = nil(A) )
             => ( Y != nil(A) ) )
           => ~ ! [X: A,Xs2: list(A)] :
                  ( ( Xa = aa(list(A),list(A),cons(A,X),Xs2) )
                 => ( Y != append(A,aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_mp(A,fun(A,$o),X)),Xs2)),append(A,aa(list(A),list(A),cons(A,X),nil(A)),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X)),Xs2)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_3761_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xs: list(A)] : aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),cons(A,Xa),Xs)) = append(A,aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_mp(A,fun(A,$o),Xa)),Xs)),append(A,aa(list(A),list(A),cons(A,Xa),nil(A)),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),Xa)),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_3762_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_3763_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_takeWhile_le
tff(fact_3764_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_3765_takeWhile__eq__take,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : takeWhile(A,P,Xs) = take(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% takeWhile_eq_take
tff(fact_3766_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))
     => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_3767_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_3768_drop__takeWhile,axiom,
    ! [A: $tType,I: nat,P: fun(A,$o),L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)))
     => ( drop(A,I,takeWhile(A,P,L)) = takeWhile(A,P,drop(A,I,L)) ) ) ).

% drop_takeWhile
tff(fact_3769_sorted__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),linorder_quicksort(A),Xs)) ) ).

% sorted_quicksort
tff(fact_3770_eq__len__takeWhile__conv,axiom,
    ! [A: $tType,I: nat,P: fun(A,$o),L: list(A)] :
      ( ( I = aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),L))
        & ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I)
           => aa(A,$o,P,aa(nat,A,nth(A,L),J3)) )
        & ( aa(nat,$o,aa(nat,fun(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)) ) ) ) ).

% eq_len_takeWhile_conv
tff(fact_3771_less__length__takeWhile__conv,axiom,
    ! [A: $tType,I: nat,P: fun(A,$o),L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
        & ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J3),I)
           => aa(A,$o,P,aa(nat,A,nth(A,L),J3)) ) ) ) ).

% less_length_takeWhile_conv
tff(fact_3772_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_3773_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(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)) )
       => ( takeWhile(A,P,Xs) = take(A,Nb,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_3774_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B),Ta: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),map(B,A,F,Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,$o),aTP_Lamp_mq(fun(B,A),fun(A,fun(B,$o)),F),Ta)),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_mq(fun(B,A),fun(A,fun(B,$o)),F),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_3775_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),Xa) = Y )
         => ( aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),Xa)
           => ( ( ( Xa = nil(A) )
               => ( ( Y = nil(A) )
                 => ~ aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),nil(A)) ) )
             => ~ ! [X: A,Xs2: list(A)] :
                    ( ( Xa = aa(list(A),list(A),cons(A,X),Xs2) )
                   => ( ( Y = append(A,aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_mp(A,fun(A,$o),X)),Xs2)),append(A,aa(list(A),list(A),cons(A,X),nil(A)),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X)),Xs2)))) )
                     => ~ aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),aa(list(A),list(A),cons(A,X),Xs2)) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_3776_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_iw(A,fun(list(A),list(A)))),Xss)) = map(list(A),A,hd(A),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_lc(list(A),$o)),Xss)) ).

% transpose_aux_filter_head
tff(fact_3777_cos__coeff__Suc,axiom,
    ! [Nb: nat] : cos_coeff(aa(nat,nat,suc,Nb)) = aa(real,real,aa(real,fun(real,real),divide_divide(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_3778_distinct__concat_H,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_lc(list(A),$o)),Xs))
     => ( ! [Ys5: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys5),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys5) )
       => ( ! [Ys5: list(A),Zs2: list(A)] :
              ( aa(set(list(A)),$o,member(list(A),Ys5),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( aa(set(list(A)),$o,member(list(A),Zs2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys5 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys5)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat'
tff(fact_3779_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_3780_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% le_inf_iff
tff(fact_3781_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_3782_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_3783_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_3784_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_3785_inf__Some,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Xa: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),aa(A,option(A),some(A),Xa)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) ) ).

% inf_Some
tff(fact_3786_Int__subset__iff,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B4) ) ) ).

% Int_subset_iff
tff(fact_3787_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(list(nat),nat,hd(nat),upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_3788_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),uminus_uminus(A,Xa)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_3789_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),uminus_uminus(A,Xa)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_3790_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),uminus_uminus(A,Xa))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_3791_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),uminus_uminus(A,Xa)),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_3792_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),uminus_uminus(A,Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_3793_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),A3)),B4) = bot_bot(set(A)) )
    <=> ( ~ aa(set(A),$o,member(A,A2),B4)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_3794_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),A3)),B4) )
    <=> ( ~ aa(set(A),$o,member(A,A2),B4)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_3795_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B4: set(A),A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(A),set(A),insert(A,A2),A3)) = bot_bot(set(A)) )
    <=> ( ~ aa(set(A),$o,member(A,A2),B4)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),A3) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_3796_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A3: set(A),B2: A,B4: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),insert(A,B2),B4)) )
    <=> ( ~ aa(set(A),$o,member(A,B2),A3)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_3797_Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B4),A3)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_3798_Compl__disjoint2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),uminus_uminus(set(A),A3)),A3) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_3799_Compl__disjoint,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),uminus_uminus(set(A),A3)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_3800_hd__replicate,axiom,
    ! [A: $tType,Nb: nat,Xa: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),replicate(A,Nb,Xa)) = Xa ) ) ).

% hd_replicate
tff(fact_3801_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3802_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F: fun(B,A),G: fun(B,A),S3: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mr(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F),G)),S3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),aa(fun(B,$o),set(B),collect(B),P)))),aa(set(B),set(A),image(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ms(fun(B,$o),fun(B,$o),P))))) ).

% if_image_distrib
tff(fact_3803_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( aa(list(A),A,hd(A),take(A,J,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_take
tff(fact_3804_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,append(A,Xs,Ys))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_3805_Union__Int__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4))) ).

% Union_Int_subset
tff(fact_3806_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ).

% Sup_inter_less_eq
tff(fact_3807_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_3808_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_sup_ord(1)
tff(fact_3809_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_le1
tff(fact_3810_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_le2
tff(fact_3811_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2) ) ) ) ).

% le_infE
tff(fact_3812_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% le_infI
tff(fact_3813_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D)) ) ) ) ).

% inf_mono
tff(fact_3814_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% le_infI1
tff(fact_3815_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xa: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% le_infI2
tff(fact_3816_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_3817_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% inf.orderI
tff(fact_3818_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X),Y3)),X)
         => ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X),Y3)),Y3)
           => ( ! [X: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F,Y3),Z3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F,Xa),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_3819_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% le_iff_inf
tff(fact_3820_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_3821_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_3822_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% inf_absorb1
tff(fact_3823_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_3824_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% inf.boundedE
tff(fact_3825_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ) ) ).

% inf.boundedI
tff(fact_3826_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ) ) ).

% inf_greatest
tff(fact_3827_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.order_iff
tff(fact_3828_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),A2) ) ).

% inf.cobounded1
tff(fact_3829_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2) ) ).

% inf.cobounded2
tff(fact_3830_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb_iff1
tff(fact_3831_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_3832_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI1
tff(fact_3833_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI2
tff(fact_3834_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),A3)
         => ! [Xa3: A] :
              ( aa(set(A),$o,member(A,Xa3),B4)
             => ( X3 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3835_Int__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_3836_Int__empty__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B4) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_3837_disjoint__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> ! [X3: A] :
          ( aa(set(A),$o,member(A,X3),A3)
         => ~ aa(set(A),$o,member(A,X3),B4) ) ) ).

% disjoint_iff
tff(fact_3838_Int__emptyI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => ~ aa(set(A),$o,member(A,X),B4) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_3839_disjointI,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A2)
         => ~ aa(set(A),$o,member(A,X),B2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A2),B2) = bot_bot(set(A)) ) ) ).

% disjointI
tff(fact_3840_inter__eq__subsetI,axiom,
    ! [A: $tType,S3: set(A),S6: set(A),A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),S6)
     => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),S6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),S6) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),S3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),S3) ) ) ) ).

% inter_eq_subsetI
tff(fact_3841_Int__Collect__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => ( aa(A,$o,P,X)
             => aa(A,$o,Q,X) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(fun(A,$o),set(A),collect(A),Q))) ) ) ).

% Int_Collect_mono
tff(fact_3842_Int__greatest,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) ) ) ).

% Int_greatest
tff(fact_3843_Int__absorb2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = A3 ) ) ).

% Int_absorb2
tff(fact_3844_Int__absorb1,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_3845_Int__lower2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),B4) ).

% Int_lower2
tff(fact_3846_Int__lower1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),A3) ).

% Int_lower1
tff(fact_3847_Int__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B4: set(A),D4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),D4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),D4)) ) ) ).

% Int_mono
tff(fact_3848_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_au(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_conj_eq
tff(fact_3849_Int__Collect,axiom,
    ! [A: $tType,Xa: A,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))
    <=> ( aa(set(A),$o,member(A,Xa),A3)
        & aa(A,$o,P,Xa) ) ) ).

% Int_Collect
tff(fact_3850_Int__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_mt(set(A),fun(set(A),fun(A,$o)),A3),B4)) ).

% Int_def
tff(fact_3851_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI2
tff(fact_3852_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI1
tff(fact_3853_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_3854_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% inf.strict_boundedE
tff(fact_3855_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_3856_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb3
tff(fact_3857_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xa: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% less_infI2
tff(fact_3858_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% less_infI1
tff(fact_3859_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z))) ) ).

% distrib_inf_le
tff(fact_3860_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z))) ) ).

% distrib_sup_le
tff(fact_3861_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),uminus_uminus(A,Xa)),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_3862_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),uminus_uminus(A,Xa)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_3863_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B4)),A2) = bot_bot(A) )
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),B4)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),A2) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3864_inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,B4: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),B4)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),inf_inf(A),A2)),B4)) ) ).

% inf_Sup
tff(fact_3865_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),B4: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,F),A3)),aa(set(B),set(A),image(B,A,F),B4))) ).

% image_Int_subset
tff(fact_3866_disjoint__mono,axiom,
    ! [A: $tType,A2: set(A),A6: set(A),B2: set(A),B6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),A6)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),B6)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6) = bot_bot(set(A)) )
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A2),B2) = bot_bot(set(A)) ) ) ) ) ).

% disjoint_mono
tff(fact_3867_Diff__triv,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),minus_minus(set(A),A3),B4) = A3 ) ) ).

% Diff_triv
tff(fact_3868_Int__Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),minus_minus(set(A),A3),B4)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_3869_disjoint__alt__simp1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A3),B4) = A3 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) ) ) ).

% disjoint_alt_simp1
tff(fact_3870_disjoint__alt__simp2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A3),B4) != A3 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp2
tff(fact_3871_insert__partition,axiom,
    ! [A: $tType,Xa: set(A),F4: set(set(A))] :
      ( ~ aa(set(set(A)),$o,member(set(A),Xa),F4)
     => ( ! [X: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X),aa(set(set(A)),set(set(A)),insert(set(A),Xa),F4))
           => ! [Xa4: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),Xa4),aa(set(set(A)),set(set(A)),insert(set(A),Xa),F4))
               => ( ( X != Xa4 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa4) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Xa),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_3872_Union__disjoint,axiom,
    ! [A: $tType,C3: set(set(A)),A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)),A3) = bot_bot(set(A)) )
    <=> ! [X3: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X3),C3)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),A3) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_3873_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3) ) ).

% Un_Int_assoc_eq
tff(fact_3874_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_3875_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),B4: set(B),A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),B4))),A2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_mu(fun(B,A),fun(A,fun(B,A)),F),A2)),B4)) ) ).

% SUP_inf
tff(fact_3876_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B4)),A2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image(A,A,aTP_Lamp_mv(A,fun(A,A),A2)),B4)) ) ).

% Sup_inf
tff(fact_3877_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F: fun(B,A),B4: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),B4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mw(A,fun(fun(B,A),fun(B,A)),A2),F)),B4)) ) ).

% inf_SUP
tff(fact_3878_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),A3: set(B),G: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_my(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B4)),A3)) ) ).

% SUP_inf_distrib2
tff(fact_3879_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),S2)),aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_Int
tff(fact_3880_Int__UN__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B),B4: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),J4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_na(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A3),B4),J4)),I5)) ).

% Int_UN_distrib2
tff(fact_3881_Int__UN__distrib,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A3: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_nb(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A3)),I5)) ).

% Int_UN_distrib
tff(fact_3882_UN__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_nc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3)) ).

% UN_extend_simps(4)
tff(fact_3883_UN__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_nb(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) ).

% UN_extend_simps(5)
tff(fact_3884_inter__set__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3)),Xs)) ).

% inter_set_filter
tff(fact_3885_Int__Union2,axiom,
    ! [A: $tType,B4: set(set(A)),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)),A3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aTP_Lamp_nd(set(A),fun(set(A),set(A)),A3)),B4)) ).

% Int_Union2
tff(fact_3886_Int__Union,axiom,
    ! [A: $tType,A3: set(A),B4: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B4)) ).

% Int_Union
tff(fact_3887_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),uminus_uminus(A,Y)) ) ) ).

% inf_shunt
tff(fact_3888_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q4: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q4),R2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),uminus_uminus(A,Q4))),R2) ) ) ).

% sup_neg_inf
tff(fact_3889_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),uminus_uminus(A,Y))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% shunt2
tff(fact_3890_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),uminus_uminus(A,Y)),Z)) ) ) ).

% shunt1
tff(fact_3891_inter__Set__filter,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = filter3(A,aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3),B4) ) ) ).

% inter_Set_filter
tff(fact_3892_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_3893_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_3894_ivl__disj__int__one_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_3895_ivl__disj__int__one_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_3896_disjoint__alt__simp3,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B4)),A3)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp3
tff(fact_3897_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),uminus_uminus(set(A),B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_3898_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(A),fun(A,B),aTP_Lamp_ne(fun(A,B),fun(set(A),fun(A,B)),G),B4)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_3899_sorted__hd__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ! [X4: A] :
                ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(list(A),A,hd(A),Xs)),X4) ) ) ) ) ).

% sorted_hd_min
tff(fact_3900_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,finite_finite2(A),S3)
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,H,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( aa(set(A),$o,member(A,I2),aa(set(A),set(A),minus_minus(set(A),S3),T2))
                   => ( aa(A,B,G,I2) = zero_zero(B) ) )
               => ( ! [X: A] :
                      ( aa(set(A),$o,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),T2))
                     => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T2) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_3901_hd__drop__conv__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),A,hd(A),drop(A,Nb,Xs)) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% hd_drop_conv_nth
tff(fact_3902_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,Xa: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),K),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_3903_sorted__hd__last,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( ( L != nil(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(list(A),A,hd(A),L)),last(A,L)) ) ) ) ).

% sorted_hd_last
tff(fact_3904_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_nf(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),uminus_uminus(set(A),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% sum.If_cases
tff(fact_3905_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B4)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_3906_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B4)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_3907_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,Nb),Xs) = aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),Xs)),take(A,Nb,aa(list(A),list(A),tl(A),Xs))) ) ) ).

% take_Suc
tff(fact_3908_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),A,hd(A),slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).

% slice_head
tff(fact_3909_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys5: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys5),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys5) )
       => ( ! [Ys5: list(A),Zs2: list(A)] :
              ( aa(set(list(A)),$o,member(list(A),Ys5),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( aa(set(list(A)),$o,member(list(A),Zs2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys5 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys5)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_3910_sum_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X)) )
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),I5)
                 => ! [Xa4: A] :
                      ( aa(set(A),$o,member(A,Xa4),I5)
                     => ( ( X != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ng(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_3911_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] :
      size_list(A,F,Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,aa(list(A),A,hd(A),Xs))),size_list(A,F,aa(list(A),list(A),tl(A),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_3912_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I: nat,J: nat] :
      ( distinct(A,Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_3913_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),A3: set(A),B4: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),S3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),$o,finite_finite2(A),B4)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = finite_fold(A,B,F,finite_fold(A,B,F,Z,A3),B4) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_3914_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),aa(list(list(A)),list(list(A)),removeAll(list(A),nil(A)),Xs))
        & ! [Ys3: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
        & ! [Ys3: list(A),Zs3: list(A)] :
            ( ( aa(set(list(A)),$o,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & aa(set(list(A)),$o,member(list(A),Zs3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & ( Ys3 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_3915_sin__coeff__Suc,axiom,
    ! [Nb: nat] : sin_coeff(aa(nat,nat,suc,Nb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos_coeff(Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ).

% sin_coeff_Suc
tff(fact_3916_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A2: A,Xs: list(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F,Xs))
           => ( ( aa(list(A),A,hd(A),aa(list(A),list(A),filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nh(A,fun(fun(A,B),fun(A,$o)),A2),F)),Xs)) = A2 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_3917_take__hd__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( append(A,take(A,Nb,Xs),aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),drop(A,Nb,Xs))),nil(A))) = take(A,aa(nat,nat,suc,Nb),Xs) ) ) ).

% take_hd_drop
tff(fact_3918_part__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B,Xs: list(A)] : linorder_part(A,B,F,Pivot,Xs) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ni(fun(A,B),fun(B,fun(A,$o)),F),Pivot)),Xs)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_nj(fun(A,B),fun(B,fun(A,$o)),F),Pivot)),Xs)),aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_nk(fun(A,B),fun(B,fun(A,$o)),F),Pivot)),Xs))) ) ).

% part_def
tff(fact_3919_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => aa(set(int),$o,member(int,signed6721504322012087516modulo(int,A2,B2)),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),uminus_uminus(int,aa(int,int,abs_abs(int),B2))),one_one(int)),aa(int,int,minus_minus(int,aa(int,int,abs_abs(int),B2)),one_one(int)))) ) ).

% smod_int_range
tff(fact_3920_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_nl(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_3921_Succ__def,axiom,
    ! [A: $tType,Kl: set(list(A)),Kl2: list(A)] : bNF_Greatest_Succ(A,Kl,Kl2) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_nm(set(list(A)),fun(list(A),fun(A,$o)),Kl),Kl2)) ).

% Succ_def
tff(fact_3922_merge__pure__and,axiom,
    ! [A2: $o,B2: $o] :
      aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn((A2))),pure_assn((B2))) = pure_assn(( (A2)
        & (B2) )) ).

% merge_pure_and
tff(fact_3923_and__extract__pure__right__ctx__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),pure_assn((B2)))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),pure_assn((B2))) ).

% and_extract_pure_right_ctx_iff
tff(fact_3924_and__extract__pure__left__ctx__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2)))),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),pure_assn((B2))) ).

% and_extract_pure_left_ctx_iff
tff(fact_3925_smod__int__0__mod,axiom,
    ! [Xa: int] : signed6721504322012087516modulo(int,zero_zero(int),Xa) = zero_zero(int) ).

% smod_int_0_mod
tff(fact_3926_smod__int__mod__0,axiom,
    ! [Xa: int] : signed6721504322012087516modulo(int,Xa,zero_zero(int)) = Xa ).

% smod_int_mod_0
tff(fact_3927_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_nn(B,A)),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3928_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_no(fun(B,nat),fun(B,A),F)),A3) ) ).

% of_nat_prod
tff(fact_3929_inf__None__2,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Xa: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),Xa),none(A)) = none(A) ) ).

% inf_None_2
tff(fact_3930_inf__None__1,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),none(A)),Y) = none(A) ) ).

% inf_None_1
tff(fact_3931_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3) = zero_zero(B) )
          <=> ? [X3: A] :
                ( aa(set(A),$o,member(A,X3),A3)
                & ( aa(A,B,F,X3) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_3932_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_3933_and__extract__pure__right__iff,axiom,
    ! [P: assn,B2: $o] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),pure_assn((B2))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),one_one(assn)),P)),pure_assn((B2))) ).

% and_extract_pure_right_iff
tff(fact_3934_and__extract__pure__left__iff,axiom,
    ! [B2: $o,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn((B2))),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),one_one(assn)),Q)),pure_assn((B2))) ).

% and_extract_pure_left_iff
tff(fact_3935_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)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_np(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(aa(set(A),$o,member(A,A2),S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_3936_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)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_nq(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(aa(set(A),$o,member(A,A2),S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_3937_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ aa(set(A),$o,member(A,Xa),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xa),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).

% prod.insert
tff(fact_3938_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.lessThan_Suc
tff(fact_3939_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atMost_Suc
tff(fact_3940_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.op_ivl_Suc
tff(fact_3941_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(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),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),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_3942_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B)),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R),S3)) ) ).

% inf_Int_eq2
tff(fact_3943_ent__conjE2,axiom,
    ! [B4: assn,C3: assn,A3: assn] :
      ( entails(B4,C3)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),B4),C3) ) ).

% ent_conjE2
tff(fact_3944_ent__conjE1,axiom,
    ! [A3: assn,C3: assn,B4: assn] :
      ( entails(A3,C3)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),B4),C3) ) ).

% ent_conjE1
tff(fact_3945_ent__conjI,axiom,
    ! [A3: assn,B4: assn,C3: assn] :
      ( entails(A3,B4)
     => ( entails(A3,C3)
       => entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),B4),C3)) ) ) ).

% ent_conjI
tff(fact_3946_norm__assertion__simps_I9_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),bot_bot(assn)),Xa) = bot_bot(assn) ).

% norm_assertion_simps(9)
tff(fact_3947_norm__assertion__simps_I10_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),bot_bot(assn)) = bot_bot(assn) ).

% norm_assertion_simps(10)
tff(fact_3948_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aTP_Lamp_kb(fun(B,A),fun(A,fun(B,A)),F),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A3),A2) ) ).

% mod_prod_eq
tff(fact_3949_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_nr(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)) ) ).

% prod_dividef
tff(fact_3950_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A3: set(B),Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A3)),Nb) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(nat,fun(B,A),aTP_Lamp_ns(fun(B,A),fun(nat,fun(B,A)),F),Nb)),A3) ) ).

% prod_power_distrib
tff(fact_3951_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B4: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_nt(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B4),G),R)),A3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_nv(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R)),B4) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3952_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [F: fun(A,B),A3: set(A)] : aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),aTP_Lamp_nw(fun(A,B),fun(A,real),F)),A3) = real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)) ) ).

% prod_norm
tff(fact_3953_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_nx(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B4)),A3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(set(B),fun(C,A),aTP_Lamp_nz(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B4) ) ).

% prod.swap
tff(fact_3954_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A3))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7121269368397514597t_prod(B,real),aTP_Lamp_oa(fun(B,A),fun(B,real),F)),A3)) ) ).

% norm_prod_le
tff(fact_3955_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ob(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A3)) ) ).

% prod.distrib
tff(fact_3956_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_oc(fun(B,A),fun(B,A),F)),A3) ) ).

% abs_prod
tff(fact_3957_inf__Int__eq,axiom,
    ! [A: $tType,R: set(A),S3: set(A),X4: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),S3)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R),S3)) ) ).

% inf_Int_eq
tff(fact_3958_inf__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),B4))) ).

% inf_set_def
tff(fact_3959_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)) ) ) ).

% prod_nonneg
tff(fact_3960_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),aa(A,B,G,I2)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ).

% prod_mono
tff(fact_3961_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)) ) ) ).

% prod_pos
tff(fact_3962_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)) ) ) ).

% prod_ge_1
tff(fact_3963_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),A3)
                & ( aa(A,B,F,X4) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_3964_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),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_od(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% prod.inter_filter
tff(fact_3965_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S3) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_oe(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S3),H),G)),aa(set(A),set(C),image(A,C,G),S3)) ) ) ) ).

% prod.image_gen
tff(fact_3966_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_of(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_3967_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_of(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3968_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F: fun(B,nat),A3: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,nat),fun(B,A),aTP_Lamp_og(A,fun(fun(B,nat),fun(B,A)),C2),F)),A3) ) ).

% power_sum
tff(fact_3969_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_oh(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_3970_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_oh(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3971_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3972_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X15: A,Y1: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X15),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S3)
             => ( ! [X: B] :
                    ( aa(set(B),$o,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,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3)) ) ) ) ) ) ).

% prod.related
tff(fact_3973_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,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),set_or7035219750837199246ssThan(A,C2,D)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3974_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xa),A3)) = $ite(aa(set(A),$o,member(A,Xa),A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3))) ) ) ) ).

% prod.insert_if
tff(fact_3975_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3976_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S3: set(A),T2: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S3)),T2)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_oi(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S3),G),H)),T2) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),H),S3) ) ) ) ) ) ).

% prod.group
tff(fact_3977_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_oj(fun(A,B),fun(set(A),fun(A,B)),G),B4)),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_3978_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)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ok(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3979_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( exp(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),I5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_ol(fun(A,B),fun(A,B),F)),I5) ) ) ) ).

% exp_sum
tff(fact_3980_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_om(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.nat_diff_reindex
tff(fact_3981_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Nb,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_on(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or1337092689740270186AtMost(nat,Nb,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3982_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1962203154675924110t_prod(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ck(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ck(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),one_one(A)) ) ).

% prod.G_def
tff(fact_3983_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),signed6721504322012087516modulo(int,A2,B2)),B2) ) ) ).

% smod_int_compares(1)
tff(fact_3984_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(2)
tff(fact_3985_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed6721504322012087516modulo(int,A2,B2)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_3986_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(6)
tff(fact_3987_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed6721504322012087516modulo(int,A2,B2)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_3988_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(8)
tff(fact_3989_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( signed6721504322012087516modulo(int,A2,B2) = modulo_modulo(int,A2,B2) ) ) ) ).

% smod_mod_positive
tff(fact_3990_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( aa(set(A),$o,member(A,I),I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F,I))
             => ( ! [I2: A] :
                    ( aa(set(A),$o,member(A,I2),I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3991_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( aa(set(A),$o,member(A,I2),I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_3992_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),minus_minus(set(A),C3),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B4) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C3) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3993_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),minus_minus(set(A),C3),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),minus_minus(set(A),C3),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C3) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3994_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3995_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3996_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,H,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( aa(set(A),$o,member(A,X),S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3997_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),minus_minus(set(A),T2),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( aa(set(A),$o,member(A,X),S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3998_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,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3999_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),B4))) ) ) ) ).

% prod.Int_Diff
tff(fact_4000_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ).

% prod.union_inter
tff(fact_4001_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4002_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4003_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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(set(nat),A,aa(fun(nat,A),fun(set(nat),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_4004_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_4005_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_4006_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_4007_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))) ) ) ) ).

% prod.last_plus
tff(fact_4008_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),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_4009_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) = semiring_char_0_fact(nat,Nb) ).

% prod_Suc_fact
tff(fact_4010_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_oo(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),uminus_uminus(set(A),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% prod.If_cases
tff(fact_4011_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),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))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_of(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4012_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),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)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_of(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_4013_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),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))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_of(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_4014_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_op(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or7035219750837199246ssThan(nat,Nb,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_4015_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_oq(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_os(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% prod.nested_swap
tff(fact_4016_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_of(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4017_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),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_kx(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))) ) ).

% fact_prod
tff(fact_4018_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ot(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% prod.nat_group
tff(fact_4019_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ou(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_os(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.nested_swap'
tff(fact_4020_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),A3)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,I2)),aa(A,B,G,I2)) ) )
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_4021_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ov(fun(nat,A),fun(nat,fun(A,A)),F),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_4022_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_4023_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,member(A,Xa),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_4024_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xa),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_4025_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_4026_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),B4),A3)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ) ).

% prod.union_diff2
tff(fact_4027_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),uminus_uminus(int,B2)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(3)
tff(fact_4028_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),signed6721504322012087516modulo(int,A2,B2)),uminus_uminus(int,B2)) ) ) ).

% smod_int_compares(5)
tff(fact_4029_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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_4030_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.head_if
tff(fact_4031_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),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_Suc
tff(fact_4032_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)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ow(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S3) = $ite(aa(set(A),$o,member(A,A2),S3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_4033_prod_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X)) )
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),I5)
                 => ! [Xa4: A] :
                      ( aa(set(A),$o,member(A,Xa4),I5)
                     => ( ( X != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ox(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_4034_part__code_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B] : linorder_part(A,B,F,Pivot,nil(A)) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),nil(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A))) ) ).

% part_code(1)
tff(fact_4035_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W,I2))),one_one(real)) )
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Z),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),W),I5)))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aa(fun(A,B),fun(A,real),aTP_Lamp_oy(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_4036_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_of(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.atMost_shift
tff(fact_4037_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_on(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_4038_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),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),minus_minus(nat,Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_rev
tff(fact_4039_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_oz(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_4040_fact__eq__fact__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Nb)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_kx(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4041_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => ( ! [B3: A] :
                  ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),minus_minus(set(A),B4),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,B3)) )
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B4)) ) ) ) ) ) ).

% prod_mono2
tff(fact_4042_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A3: set(A),F: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( aa(A,B,F,A2) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(A,B,F,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)) ) ) ) ) ).

% prod_diff1
tff(fact_4043_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A3: set(A),B4: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B4)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
                 => ( aa(A,B,F,X) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ) ) ).

% prod_Un
tff(fact_4044_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_oz(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_4045_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_pa(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_4046_fact__div__fact,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Nb)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_kx(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_4047_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),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_4048_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_pb(nat,fun(nat,fun(nat,A)),K),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4049_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_pc(A,fun(nat,fun(nat,A)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4050_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_pa(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_4051_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)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_pd(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_4052_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)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_pd(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_4053_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_nl(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4054_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,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_pe(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_pf(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4055_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xa,Xaa) = Y )
     => ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),cons(int,Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) )
           => ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_4056_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),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_4057_Pow__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( pow(A,A3) = finite_fold(A,set(set(A)),aTP_Lamp_pg(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),insert(set(A),bot_bot(set(A))),bot_bot(set(set(A)))),A3) ) ) ).

% Pow_fold
tff(fact_4058_Pow__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(set(A)),$o,member(set(A),A3),pow(A,B4))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ).

% Pow_iff
tff(fact_4059_PowI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => aa(set(set(A)),$o,member(set(A),A3),pow(A,B4)) ) ).

% PowI
tff(fact_4060_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_4061_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A3))
      <=> ! [X3: A] :
            ( aa(set(A),$o,member(A,X3),A3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4062_Pow__empty,axiom,
    ! [A: $tType] : pow(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),insert(set(A),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_4063_Pow__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),Y6: set(A)] :
      ( ( pow(A,X5) = aa(set(set(A)),set(set(A)),insert(set(A),Y6),bot_bot(set(set(A)))) )
    <=> ( ( X5 = bot_bot(set(A)) )
        & ( Y6 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_4064_less__eq__assn__def,axiom,
    ! [A2: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A2),B2)
    <=> ( A2 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A2),B2) ) ) ).

% less_eq_assn_def
tff(fact_4065_minus__assn__def,axiom,
    ! [A2: assn,B2: assn] : aa(assn,assn,minus_minus(assn,A2),B2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A2),uminus_uminus(assn,B2)) ).

% minus_assn_def
tff(fact_4066_assn__aci_I4_J,axiom,
    ! [Xa: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y) ).

% assn_aci(4)
tff(fact_4067_assn__aci_I3_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Z)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Z)) ).

% assn_aci(3)
tff(fact_4068_assn__aci_I1_J,axiom,
    ! [Xa: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Xa) ).

% assn_aci(1)
tff(fact_4069_norm__assertion__simps_I31_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Xa) = Xa ).

% norm_assertion_simps(31)
tff(fact_4070_norm__assertion__simps_I14_J,axiom,
    ! [Xa: assn,Y: assn,Z: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),Y)),Z) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Z)) ).

% norm_assertion_simps(14)
tff(fact_4071_int__prod,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A3)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7121269368397514597t_prod(A,int),aTP_Lamp_hu(fun(A,nat),fun(A,int),F)),A3) ).

% int_prod
tff(fact_4072_Pow__bottom,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(set(A)),$o,member(set(A),bot_bot(set(A))),pow(A,B4)) ).

% Pow_bottom
tff(fact_4073_PowD,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(set(A)),$o,member(set(A),A3),pow(A,B4))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ).

% PowD
tff(fact_4074_sorted__upto,axiom,
    ! [M: int,Nb: int] : sorted_wrt(int,ord_less_eq(int),upto(M,Nb)) ).

% sorted_upto
tff(fact_4075_Pow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : pow(A,A3) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_4076_Pow__def,axiom,
    ! [A: $tType,A3: set(A)] : pow(A,A3) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_at(set(A),fun(set(A),$o),A3)) ).

% Pow_def
tff(fact_4077_Pow__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pow(A,A3)),pow(A,B4)) ) ).

% Pow_mono
tff(fact_4078_Un__Pow__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A3)),pow(A,B4))),pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))) ).

% Un_Pow_subset
tff(fact_4079_subset__Pow__Union,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A3),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3))) ).

% subset_Pow_Union
tff(fact_4080_Fpow__subset__Pow,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),pow(A,A3)) ).

% Fpow_subset_Pow
tff(fact_4081_prod__int__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_ph(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_4082_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_4083_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),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_4084_upto_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xa,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),cons(int,Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_4085_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = 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_4086_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_4087_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),aa(list(int),list(int),cons(int,J),nil(int))) ) ) ).

% upto_rec2
tff(fact_4088_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_pi(fun(B,set(A)),fun(B,set(set(A))),B4)),A3))),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))) ).

% UN_Pow_subset
tff(fact_4089_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),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))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_ph(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_4090_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = 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_4091_Fpow__Pow__finite,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,A3)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_4092_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A3: set(B),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F),A3)),B4)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F)),pow(B,A3))),pow(A,B4)) ) ).

% image_Pow_mono
tff(fact_4093_Pow__set_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      pow(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xa),Xs))) = $let(
        a2: set(set(A)),
        a2:= pow(A,aa(list(A),set(A),set2(A),Xs)),
        aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a2),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,Xa)),a2)) ) ).

% Pow_set(2)
tff(fact_4094_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F,I2)) )
       => ( ln_ln(real,aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F),I5)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_pj(fun(A,real),fun(A,real),F)),I5) ) ) ) ).

% ln_prod
tff(fact_4095_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),insert(set(A),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_4096_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),M),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_4097_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Nb: num] :
      upto(uminus_uminus(int,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Nb)),aa(list(int),list(int),cons(int,uminus_uminus(int,aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(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_4098_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Nb: num] :
      upto(uminus_uminus(int,aa(num,int,numeral_numeral(int),M)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),M))),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb))),aa(list(int),list(int),cons(int,uminus_uminus(int,aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),M))),one_one(int)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_4099_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb)),aa(list(int),list(int),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_4100_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ) ).

% numeral_le_iff
tff(fact_4101_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Nb) ) ) ).

% numeral_less_iff
tff(fact_4102_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_4103_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Va2: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Va2)),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),Va2),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_4104_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [Va2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Va2)),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),Va2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Va2)),C2)) ) ).

% distrib_left_numeral
tff(fact_4105_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,Va2: 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),Va2)) = 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),Va2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),Va2))) ) ).

% distrib_right_numeral
tff(fact_4106_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),M) ) ) ).

% neg_numeral_le_iff
tff(fact_4107_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Nb),M) ) ) ).

% neg_numeral_less_iff
tff(fact_4108_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [Va2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Va2)),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),Va2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Va2)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_4109_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,Va2: 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),Va2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),Va2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),Va2))) ) ).

% left_diff_distrib_numeral
tff(fact_4110_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Va2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Va2),W)))),Y) ) ).

% semiring_norm(168)
tff(fact_4111_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),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),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_4112_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),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Nb)) = 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_4113_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)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))) = 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_4114_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Va2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),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),uminus_uminus(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Va2),W)))),Y) ) ).

% semiring_norm(170)
tff(fact_4115_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Va2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Va2)),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Va2),W)))),Y) ) ).

% semiring_norm(171)
tff(fact_4116_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Va2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),aa(A,A,aa(A,fun(A,A),times_times(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),Va2),W))),Y) ) ).

% semiring_norm(172)
tff(fact_4117_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(4)
tff(fact_4118_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(3)
tff(fact_4119_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),Va2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),Va2)),aa(num,A,numeral_numeral(A),Va2),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_4120_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_4121_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) = A2 )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_4122_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_4123_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_4124_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(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_4125_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(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_4126_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2)),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_4127_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),Va2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),Va2)),aa(num,A,numeral_numeral(A),Va2),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_4128_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_4129_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_4130_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_4131_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))) = A2 )
        <=> $ite(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),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4132_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))) )
        <=> $ite(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),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4133_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4134_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4135_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4136_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4137_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Nb)),Xa) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4138_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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_4139_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Nb)),Xa) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4140_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),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_4141_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_4142_less__assn__def,axiom,
    ! [A2: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less(assn),A2),B2)
    <=> ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A2),B2)
        & ( A2 != B2 ) ) ) ).

% less_assn_def
tff(fact_4143_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_4144_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_le_numeral
tff(fact_4145_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_4146_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_less_numeral
tff(fact_4147_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_le_numeral
tff(fact_4148_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_4149_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : zero_zero(A) != uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_neq_neg_numeral
tff(fact_4150_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_le_numeral
tff(fact_4151_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_le_neg_numeral
tff(fact_4152_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_less_numeral
tff(fact_4153_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_less_neg_numeral
tff(fact_4154_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,Xa: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),uminus_uminus(A,Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_4155_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_le_neg_numeral
tff(fact_4156_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_4157_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_less_neg_numeral
tff(fact_4158_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_4159_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4160_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2),aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4161_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_4162_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_4163_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),uminus_uminus(A,one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_4164_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),uminus_uminus(A,one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_4165_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_4166_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,one_one(A))),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_4167_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_less_neg_numeral
tff(fact_4168_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),uminus_uminus(A,one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_4169_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_less_numeral
tff(fact_4170_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_4171_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4172_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4173_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( uminus_uminus(A,aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2) = B2,uminus_uminus(A,aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4174_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = 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),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4175_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4176_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4177_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4178_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4179_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),uminus_uminus(A,aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4180_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4181_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Xa))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Xa))),Nb)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4182_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Xa))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Xa))),Nb)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4183_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Xa))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Xa))),Nb)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4184_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Xa))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Xa))),Nb)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4185_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_4186_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_4187_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_4188_word__gt__0__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( aa(word(A),$o,aa(word(A),fun(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_4189_of__int__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & ring_1(A) )
     => ! [Nb: nat] : aa(int,A,ring_1_of_int(A),semiring_char_0_fact(int,Nb)) = semiring_char_0_fact(A,Nb) ) ).

% of_int_fact
tff(fact_4190_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_4191_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_4192_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_4193_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_4194_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% of_int_le_iff
tff(fact_4195_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(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,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% of_int_less_iff
tff(fact_4196_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_4197_word__of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Bin))) = uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),Bin)) ) ).

% word_of_int_neg_numeral
tff(fact_4198_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_pk(fun(B,int),fun(B,A),F)),A3) ) ).

% of_int_prod
tff(fact_4199_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_4200_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_4201_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_pl(fun(B,int),fun(B,A),F)),A3) ) ).

% of_int_sum
tff(fact_4202_word__of__int__neg__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),uminus_uminus(int,one_one(int))) = uminus_uminus(word(A),one_one(word(A))) ) ) ).

% word_of_int_neg_1
tff(fact_4203_word__le__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A)))) ) ) ) ).

% word_le_sub1_numberof
tff(fact_4204_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_4205_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_4206_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_4207_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_le_numeral_iff
tff(fact_4208_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_4209_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_4210_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_4211_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_less_numeral_iff
tff(fact_4212_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_4213_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_4214_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_4215_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_4216_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_4217_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_4218_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xa) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4219_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4220_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xa) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4221_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4222_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_4223_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(num,real,numeral_numeral(real),W))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_4224_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,M: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Nb)),aa(nat,real,semiring_1_of_nat(real),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Nb)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_4225_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,aa(nat,fun(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_4226_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4227_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4228_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4229_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4230_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_4231_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_le_of_int
tff(fact_4232_pochhammer__of__int,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: int,Nb: nat] : comm_s3205402744901411588hammer(A,aa(int,A,ring_1_of_int(A),Xa),Nb) = aa(int,A,ring_1_of_int(A),comm_s3205402744901411588hammer(int,Xa,Nb)) ) ).

% pochhammer_of_int
tff(fact_4233_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,Nb: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),Nb) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Nb)) ) ).

% word_of_int_power_hom
tff(fact_4234_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xa)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xa)) ) ).

% mult_of_int_commute
tff(fact_4235_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_less_of_int
tff(fact_4236_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),Xa) ) ).

% ex_of_int_less
tff(fact_4237_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2))) ) ).

% word_neg_numeral_alt
tff(fact_4238_wi__hom__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),B2)) ) ).

% wi_hom_add
tff(fact_4239_wi__hom__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int] : uminus_uminus(word(A),aa(int,word(A),ring_1_of_int(word(A)),A2)) = aa(int,word(A),ring_1_of_int(word(A)),uminus_uminus(int,A2)) ) ).

% wi_hom_neg
tff(fact_4240_wi__hom__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),minus_minus(word(A),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,A2),B2)) ) ).

% wi_hom_sub
tff(fact_4241_wi__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% wi_hom_mult
tff(fact_4242_real__of__int__div4,axiom,
    ! [Nb: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xa))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Nb)),aa(int,real,ring_1_of_int(real),Xa))) ).

% real_of_int_div4
tff(fact_4243_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_4244_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_4245_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xa)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% of_int_leD
tff(fact_4246_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xa)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% of_int_lessD
tff(fact_4247_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int))))
          & ! [Y4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),Xa)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int)))) )
             => ( Y4 = X ) ) ) ) ).

% floor_exists1
tff(fact_4248_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int)))) ) ) ).

% floor_exists
tff(fact_4249_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Nb)),Xa) ) ) ).

% of_nat_less_of_int_iff
tff(fact_4250_int__le__real__less,axiom,
    ! [Nb: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),M)
    <=> aa(real,$o,aa(real,fun(real,$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_4251_int__less__real__le,axiom,
    ! [Nb: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_4252_real__of__int__div2,axiom,
    ! [Nb: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Nb)),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xa)))) ).

% real_of_int_div2
tff(fact_4253_real__of__int__div3,axiom,
    ! [Nb: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Nb)),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xa)))),one_one(real)) ).

% real_of_int_div3
tff(fact_4254_enat__ord__number_I1_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(1)
tff(fact_4255_enat__ord__number_I2_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(2)
tff(fact_4256_mintlistlength,axiom,
    ! [Mi: 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Nb)
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Ma)
          & ? [M2: nat] :
              ( ( aa(nat,option(nat),some(nat),M2) = vEBT_vebt_mint(Summary) )
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_4257_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),one_one(A))),Xa) ) ) ).

% neg_numeral_le_ceiling
tff(fact_4258_semiring__norm_I87_J,axiom,
    ! [M: num,Nb: num] :
      ( ( bit0(M) = bit0(Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(87)
tff(fact_4259_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb)
    <=> ( Nb != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_4260_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2) ).

% pow_sum
tff(fact_4261_two__realpow__ge__two,axiom,
    ! [Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Nb))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)) ) ).

% two_realpow_ge_two
tff(fact_4262_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Nb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
     => ( vEBT_invar_vebt(Tree,Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ).

% member_bound
tff(fact_4263_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one2 != bit0(Nb) ).

% semiring_norm(83)
tff(fact_4264_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_4265_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,M: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(Ta) )
       => aa(nat,$o,aa(nat,fun(nat,$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_4266_two__powr__height__bound__deg,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% two_powr_height_bound_deg
tff(fact_4267_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_4268_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$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_4269_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$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_4270_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_4271_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_4272_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(Ta,Xa)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
              | ( Xa = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_4273_count__buildup,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb))) ).

% count_buildup
tff(fact_4274_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_4275_ceiling__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_ceiling(A,aa(int,A,ring_1_of_int(A),Z)) = Z ) ).

% ceiling_of_int
tff(fact_4276_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_4277_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_4278_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_4279_delt__out__of__range,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_4280_semiring__norm_I78_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Nb) ) ).

% semiring_norm(78)
tff(fact_4281_semiring__norm_I71_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(71)
tff(fact_4282_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),one2) ).

% semiring_norm(75)
tff(fact_4283_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),one2),Nb) ).

% semiring_norm(68)
tff(fact_4284_del__single__cont,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & ( Xa = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ) ) ) ).

% del_single_cont
tff(fact_4285_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_4286_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),Xa)),one_one(nat)) ) ).

% tdeletemimi'
tff(fact_4287_count__buildup_H,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) ).

% count_buildup'
tff(fact_4288_mi__ma__2__deg,axiom,
    ! [Mi: 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$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_4289_succ__min,axiom,
    ! [Deg: nat,Xa: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = aa(nat,option(nat),some(nat),Mi) ) ) ) ).

% succ_min
tff(fact_4290_pred__max,axiom,
    ! [Deg: nat,Ma: nat,Xa: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = aa(nat,option(nat),some(nat),Ma) ) ) ) ).

% pred_max
tff(fact_4291_cnt__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)),one_one(real)))) ) ).

% cnt_bound'
tff(fact_4292_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_4293_TBOUND__vebt__memberi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xa: nat] : time_TBOUND($o,vEBT_V854960066525838166emberi(Ta,Tib,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)))) ).

% TBOUND_vebt_memberi
tff(fact_4294_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(Ta)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),one_one(nat)))) ) ).

% inrange
tff(fact_4295_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_4296_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_4297_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),Va2)) = aa(num,int,numeral_numeral(int),Va2) ) ).

% ceiling_numeral
tff(fact_4298_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_4299_ceiling__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: nat] : archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ).

% ceiling_of_nat
tff(fact_4300_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_4301_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit0(Nb)) ).

% semiring_norm(76)
tff(fact_4302_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_4303_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),Z) ) ).

% ceiling_add_of_int
tff(fact_4304_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xa),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xa)),Z) ) ).

% ceiling_diff_of_int
tff(fact_4305_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,one_one(A))),uminus_uminus(A,aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4306_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))),uminus_uminus(A,one_one(A)))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4307_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_4308_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_4309_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_4310_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_4311_div2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% div2_Suc_Suc
tff(fact_4312_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_4313_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_4314_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_4315_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),Va2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(num,A,numeral_numeral(A),Va2)) ) ) ).

% ceiling_le_numeral
tff(fact_4316_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% zero_less_ceiling
tff(fact_4317_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Va2)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Va2)),Xa) ) ) ).

% numeral_less_ceiling
tff(fact_4318_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_4319_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% one_le_ceiling
tff(fact_4320_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(num,A,numeral_numeral(A),Va2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),Va2)) ) ).

% ceiling_add_numeral
tff(fact_4321_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_4322_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ).

% one_less_ceiling
tff(fact_4323_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num] : archimedean_ceiling(A,uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) = uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2)) ) ).

% ceiling_neg_numeral
tff(fact_4324_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_4325_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xa),aa(num,A,numeral_numeral(A),Va2))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),Va2)) ) ).

% ceiling_diff_numeral
tff(fact_4326_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),one2) ) ) ).

% numeral_le_one_iff
tff(fact_4327_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Nb) ) ) ).

% one_less_numeral_iff
tff(fact_4328_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xa),one_one(A))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xa)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_4329_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_4330_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_4331_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_4332_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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))) )
            <=> ( Xa = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_4333_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(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_4334_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: 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),Xa),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) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_4335_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_4336_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_4337_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),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_4338_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_4339_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_4340_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_4341_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_4342_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),uminus_uminus(A,one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_4343_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,one_one(A))),Xa) ) ) ).

% zero_le_ceiling
tff(fact_4344_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(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_4345_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_4346_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),Va2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),Va2)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_4347_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Va2)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),Va2)),one_one(A))),Xa) ) ) ).

% numeral_le_ceiling
tff(fact_4348_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_4349_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),Xa) ) ) ).

% neg_numeral_less_ceiling
tff(fact_4350_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,minus_minus(A,uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_4351_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_4352_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_4353_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xa: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Z),Y)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),aa(extended_enat,extended_enat,minus_minus(extended_enat,Y),Z)) = aa(extended_enat,extended_enat,minus_minus(extended_enat,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_4354_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Nb),zero_zero(extended_enat))
    <=> ( Nb = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_4355_i0__lb,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),zero_zero(extended_enat)),Nb) ).

% i0_lb
tff(fact_4356_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$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,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_4357_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_4358_word__of__int__Ex,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
        ? [Y3: int] : Xa = aa(int,word(A),ring_1_of_int(word(A)),Y3) ) ).

% word_of_int_Ex
tff(fact_4359_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_4360_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_4361_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_4362_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_4363_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_4364_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_4365_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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),Xa),Xa)),Xa)),Xa) ) ).

% power4_eq_xxxx
tff(fact_4366_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_4367_pos2,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_4368_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_4369_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_4370_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_4371_n__less__equal__power__2,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$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_4372_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xa),one2)
    <=> ( Xa = one2 ) ) ).

% le_num_One_iff
tff(fact_4373_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)) ) ).

% self_le_ge2_pow
tff(fact_4374_power2__nat__le__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_4375_power2__nat__le__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_4376_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ).

% le_of_int_ceiling
tff(fact_4377_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xa)) ) ) ).

% ceiling_mono
tff(fact_4378_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% ceiling_less_cancel
tff(fact_4379_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_4380_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_4381_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_4382_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_4383_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( Xa = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_4384_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_4385_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(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_4386_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_4387_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_4388_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(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_4389_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_4390_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_4391_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_4392_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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_4393_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_4394_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
     => ? [A4: A,B3: A] : L = aa(list(A),list(A),cons(A,A4),aa(list(A),list(A),cons(A,B3),nil(A))) ) ).

% list_decomp_2
tff(fact_4395_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Nb)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_4396_realpow__square__minus__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_4397_pos__mod__sign2,axiom,
    ! [A2: int] : aa(int,$o,aa(int,fun(int,$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_4398_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_4399_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_4400_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_4401_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_4402_binomial__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_4403_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)) ).

% two_realpow_ge_one
tff(fact_4404_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_4405_binomial__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_mono
tff(fact_4406_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% binomial_le_pow2
tff(fact_4407_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(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_4408_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_4409_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_4410_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_4411_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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))
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4412_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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_4413_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$o,aa(A,fun(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),Xa),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_4414_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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),Xa),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)))))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_4415_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(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_4416_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: 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),Xa),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),Xa),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))),Xa)),Y)) ) ).

% power2_sum
tff(fact_4417_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_4418_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_4419_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% zero_le_even_power'
tff(fact_4420_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xa: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$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,aa(A,A,abs_abs(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_4421_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_4422_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_4423_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_4424_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_4425_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_4426_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_4427_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_4428_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),uminus_uminus(A,A2)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(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_4429_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,aa(nat,fun(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_4430_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_4431_two__pow__div__gt__le,axiom,
    ! [Va2: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Va2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% two_pow_div_gt_le
tff(fact_4432_nat__add__offset__less,axiom,
    ! [Y: nat,Nb: nat,Xa: nat,M: nat,Sz: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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,$o,aa(nat,fun(nat,$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),Xa),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_4433_nat__power__less__diff,axiom,
    ! [Nb: nat,Q4: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Q4),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_4434_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_4435_nat__le__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_le_power_trans
tff(fact_4436_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,aa(nat,fun(nat,$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_4437_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_4438_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_strict_mono
tff(fact_4439_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_4440_axxmod2,axiom,
    ! [Xa: 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)),Xa)),Xa),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)),Xa)),Xa),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% axxmod2
tff(fact_4441_axxdiv2,axiom,
    ! [Xa: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Xa)),Xa)),aa(num,int,numeral_numeral(int),bit0(one2))) = Xa )
      & ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),Xa)),Xa)),aa(num,int,numeral_numeral(int),bit0(one2))) = Xa ) ) ).

% axxdiv2
tff(fact_4442_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,Nb)),semiring_char_0_fact(real,Nb))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% square_fact_le_2_fact
tff(fact_4443_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% exp_half_le2
tff(fact_4444_triangle__def,axiom,
    ! [Nb: nat] : nat_triangle(Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_4445_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_4446_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),A2) ) ) ).

% ceiling_le
tff(fact_4447_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% less_ceiling_iff
tff(fact_4448_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_4449_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_4450_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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_4451_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4452_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,Xa),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),Xa),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))),Xa)),Y)) ) ).

% power2_diff
tff(fact_4453_choose__row__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(Nb)),aa(nat,set(nat),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_4454_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_4455_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(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_4456_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(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))) = uminus_uminus(A,one_one(A)) ) ).

% power_minus1_odd
tff(fact_4457_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( 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_4458_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$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_4459_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_4460_less__two__pow__divD,axiom,
    ! [Xa: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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_4461_less__two__pow__divI,axiom,
    ! [Xa: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% less_two_pow_divI
tff(fact_4462_nat__less__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$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_4463_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q4: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Nb)),aa(num,A,numeral_numeral(A),bit0(Q4))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4464_choose__two,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_4465_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,uminus_uminus(int,one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$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 != 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_4466_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_4467_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),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_4468_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_4469_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_as(nat,fun(nat,$o)),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_4470_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),uminus_uminus(A,aa(num,A,numeral_numeral(A),one2))) = uminus_uminus(A,B2) ) ).

% mult_1s_ring_1(2)
tff(fact_4471_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),uminus_uminus(A,aa(num,A,numeral_numeral(A),one2))),B2) = uminus_uminus(A,B2) ) ).

% mult_1s_ring_1(1)
tff(fact_4472_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_4473_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_4474_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_pm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.in_pairs
tff(fact_4475_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_pm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.in_pairs_0
tff(fact_4476_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_pn(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.in_pairs
tff(fact_4477_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),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,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_pn(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.in_pairs_0
tff(fact_4478_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( ( modulo_modulo(A,Xa,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = modulo_modulo(A,Xa,M) )
              | ( modulo_modulo(A,Xa,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,Xa,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_4479_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(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_4480_choose__square__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_po(nat,fun(nat,nat),Nb)),aa(nat,set(nat),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_4481_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_4482_nat__div__eq__Suc__0__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & aa(nat,$o,aa(nat,fun(nat,$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_4483_power__2__mult__step__le,axiom,
    ! [N3: nat,Nb: nat,K7: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$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)),K7)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),K))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K7),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_4484_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(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_4485_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2) ) ).

% of_int_ceiling_diff_one_le
tff(fact_4486_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( 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_4487_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_4488_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_4489_exp__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),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)),Xa)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_4490_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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),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_4491_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,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_as(nat,fun(nat,$o)),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_4492_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_kx(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_4493_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(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_4494_Sum__Ico__nat,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_kx(nat,nat)),set_or7035219750837199246ssThan(nat,M,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_4495_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xa: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_4496_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),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_4497_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),uminus_uminus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),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_4498_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( 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_4499_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),uminus_uminus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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),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_4500_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_4501_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),ln_ln(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_4502_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),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))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_4503_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,aa(fun(nat,A),fun(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_4504_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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_pp(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_4505_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),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),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),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_4506_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_pq(nat,fun(nat,A),M)),aa(nat,set(nat),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_4507_arith__series__nat,axiom,
    ! [A2: nat,D: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_pr(nat,fun(nat,fun(nat,nat)),A2),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),D)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_4508_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4509_Sum__Icc__nat,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_kx(nat,nat)),set_or1337092689740270186AtMost(nat,M,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_4510_Suc__nat__number__of__add,axiom,
    ! [Va2: num,Nb: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),Va2)),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),Va2),one2))),Nb) ).

% Suc_nat_number_of_add
tff(fact_4511_Sum__Icc__int,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),Nb)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_ph(int,int)),set_or1337092689740270186AtMost(int,M,Nb)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_4512_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(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)))),aa(nat,set(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_4513_choose__linear__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ps(nat,fun(nat,nat),Nb)),aa(nat,set(nat),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_4514_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I3)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I3)) )
             => aa(int,$o,P,I3) ) ) ) ).

% ceiling_split
tff(fact_4515_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( ( archimedean_ceiling(A,Xa) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_4516_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xa) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_4517_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))),one_one(A))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ) ).

% ceiling_correct
tff(fact_4518_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2))) ) ) ) ).

% mult_ceiling_le
tff(fact_4519_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_4520_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% le_ceiling_iff
tff(fact_4521_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))),Nb))),semiring_char_0_fact(A,Nb)) ) ).

% fact_double
tff(fact_4522_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))))),exp(real,Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_4523_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,aa(fun(nat,A),fun(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_4524_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_4525_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D: A,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_pt(A,fun(A,fun(nat,A)),A2),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_4526_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,ln_ln(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_4527_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
               => ( ! [X: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_4528_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),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),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),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_4529_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),comm_s3205402744901411588hammer(A,Z,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)) ) ).

% pochhammer_double
tff(fact_4530_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_4531_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa(nat,nat,binomial(Nb),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,minus_minus(nat,Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Nb),K)),one_one(nat)),Nb,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_4532_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_pu(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4533_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Nb) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
               => ( ! [X: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_4534_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$o,aa(A,fun(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,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q4)))),Q4)) ) ) ).

% ceiling_divide_upper
tff(fact_4535_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),Nb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Nb)))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_4536_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,uminus_uminus(real,Xa)),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),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),ln_ln(real,aa(real,real,minus_minus(real,one_one(real)),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_4537_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,ln_ln(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),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),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_4538_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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)))),aa(nat,set(nat),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_4539_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q4)))),one_one(A))),Q4)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_4540_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_pu(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4541_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Nb)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),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,Xa) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_4542_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_pw(A,fun(A,fun(nat,fun(nat,A))),H),Z),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_4543_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,ln_ln(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),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),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_4544_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_px(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_4545_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K4: real,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K4),aa(nat,nat,minus_minus(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_4546_Tb__T__vebt__buildupi,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Nb))),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Nb)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi
tff(fact_4547_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,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
              uminus_uminus(A,aa(int,A,ring_1_of_int(A),uminus_uminus(int,K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(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_4548_Tb__T__vebt__buildupi_H,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),vEBT_V9176841429113362141ildupi(Nb)),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Nb)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi'
tff(fact_4549_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Nb)),aa(nat,nat,minus_minus(nat,vEBT_VEBT_Tb2(Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Tb_T_vebt_buildupi''
tff(fact_4550_Tb__Tb_H,axiom,
    ! [Ta: nat] : vEBT_VEBT_Tb(Ta) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(Ta)) ).

% Tb_Tb'
tff(fact_4551_max__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q4) = Q4 ).

% max_enat_simps(3)
tff(fact_4552_max__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q4),zero_zero(extended_enat)) = Q4 ).

% max_enat_simps(2)
tff(fact_4553_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_4554_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_4555_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_4556_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_4557_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_4558_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_4559_div__half__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),Y)),modulo_modulo(nat,Xa,Y)) = $let(
            q: nat,
            q:= aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)),
            $let(
              r: nat,
              r:= aa(nat,nat,minus_minus(nat,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),q),Y)),
              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),r),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),q),one_one(nat))),aa(nat,nat,minus_minus(nat,r),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_4560_htt__vebt__memberi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xa: nat] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_memberi(Tib,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_ag(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Ta),Tib),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)))) ).

% htt_vebt_memberi
tff(fact_4561_vebt__buildupi__rule,axiom,
    ! [Nb: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% vebt_buildupi_rule
tff(fact_4562_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q4: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q4),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q4)),one_one(A))),aa(A,A,minus_minus(A,R2),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q4)),R2)) ) ).

% divmod_step_eq
tff(fact_4563_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_4564_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_4565_semiring__norm_I89_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,bit1,M) != bit0(Nb) ).

% semiring_norm(89)
tff(fact_4566_semiring__norm_I88_J,axiom,
    ! [M: num,Nb: num] : bit0(M) != aa(num,num,bit1,Nb) ).

% semiring_norm(88)
tff(fact_4567_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_4568_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit1,Nb) ).

% semiring_norm(84)
tff(fact_4569_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ).

% tdeletemimi
tff(fact_4570_semiring__norm_I80_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Nb) ) ).

% semiring_norm(80)
tff(fact_4571_semiring__norm_I73_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(73)
tff(fact_4572_minNull__delete__time__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(Ta,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(Ta,Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% minNull_delete_time_bound
tff(fact_4573_Tb_H__cnt,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_VEBT_Tb2(Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_vebt_buildup(Nb)))) ).

% Tb'_cnt
tff(fact_4574_T__vebt__buildupi__cnt_H,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V441764108873111860ildupi(Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb)))) ).

% T_vebt_buildupi_cnt'
tff(fact_4575_TBOUND__buildupi,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => time_TBOUND(vEBT_VEBTi,vEBT_vebt_buildupi(Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% TBOUND_buildupi
tff(fact_4576_delete__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))) ) ).

% delete_bound_height
tff(fact_4577_cnt__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))))))) ) ).

% cnt_bound
tff(fact_4578_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_4579_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_4580_TBOUND__vebt__inserti,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xa: nat] :
      time_TBOUND(vEBT_VEBTi,vEBT_V3964819847710782039nserti(Ta,Tib,Xa),
        $ite(vEBT_VEBT_minNull(Ta),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))))) ).

% TBOUND_vebt_inserti
tff(fact_4581_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_4582_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_4583_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_4584_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_4585_semiring__norm_I72_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(72)
tff(fact_4586_semiring__norm_I81_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,M)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Nb) ) ).

% semiring_norm(81)
tff(fact_4587_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),one2) ).

% semiring_norm(70)
tff(fact_4588_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),aa(num,num,bit1,Nb)) ).

% semiring_norm(77)
tff(fact_4589_htt__vebt__inserti,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xa: nat] : time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_inserti(Tib,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)))) ).

% htt_vebt_inserti
tff(fact_4590_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_4591_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_4592_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_4593_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_4594_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_4595_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_4596_semiring__norm_I74_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Nb) ) ).

% semiring_norm(74)
tff(fact_4597_semiring__norm_I79_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(79)
tff(fact_4598_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,Va2: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),Va2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),Va2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_4599_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_4600_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,Va2: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),Va2)) = 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),Va2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4601_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_4602_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_4603_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_4604_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_4605_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q4: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),bit0(Q4))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_4606_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_4607_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_4608_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_4609_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb2(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_4610_maxt__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_a_x_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_4611_VEBT__internal_OTb_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb(zero_zero(nat)) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_4612_mint__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_4613_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_4614_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q4: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q4))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q4))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4615_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q4: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q4))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),bit0(Q4))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4616_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_4617_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_4618_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_4619_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb2(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_4620_VEBT__internal_OTb_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb(aa(nat,nat,suc,zero_zero(nat))) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_4621_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_4622_small__powers__of__2,axiom,
    ! [Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Xa)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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,Xa),one_one(nat)))) ) ).

% small_powers_of_2
tff(fact_4623_pred__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))) ) ).

% pred_bound_height
tff(fact_4624_succ__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))) ) ).

% succ_bound_height
tff(fact_4625_space__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Ta)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% space_bound
tff(fact_4626_space__2__pow__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,one2))))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)),one_one(real)))) ) ).

% space_2_pow_bound
tff(fact_4627_space__cnt,axiom,
    ! [Ta: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta))) ).

% space_cnt
tff(fact_4628_space__space_H,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Ta)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta)) ).

% space_space'
tff(fact_4629_space_H__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% space'_bound
tff(fact_4630_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),set(code_integer),image(code_integer,code_integer,aTP_Lamp_py(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_4631_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_4632_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_4633_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_4634_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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList),zero_zero(nat))) ).

% VEBT_internal.space'.simps(2)
tff(fact_4635_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(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] :
              ( ( Xa = 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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_4636_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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList),zero_zero(nat))) ).

% VEBT_internal.space.simps(2)
tff(fact_4637_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(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] :
              ( ( Xa = 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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_4638_t__build__cnt,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_build_cnt
tff(fact_4639_t__buildup__cnt,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8346862874174094_d_u_p(Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_buildup_cnt
tff(fact_4640_vebt__buildup__bound,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V8346862874174094_d_u_p(Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_4641_buildup__build__time,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_V8346862874174094_d_u_p(Nb)),vEBT_V8646137997579335489_i_l_d(Nb)) ).

% buildup_build_time
tff(fact_4642_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_4643_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_4644_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_4645_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_4646_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = 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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_4647_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),Xa)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = 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))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_4648_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K2: int,L2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L2))
           => ( ( ~ ( aa(set(int),$o,member(int,K2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int)))))
                    & aa(set(int),$o,member(int,L2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_4649_setceilmax,axiom,
    ! [S2: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),Nb: nat] :
      ( vEBT_invar_vebt(S2,M)
     => ( ! [X: vEBT_VEBT] :
            ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
           => vEBT_invar_vebt(X,Nb) )
       => ( ( M = aa(nat,nat,suc,Nb) )
         => ( ! [X: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,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,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,S2)) = archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))) )
             => ( aa(nat,int,semiring_1_of_nat(int),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,S2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))))) = archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_4650_pred__list__to__short,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_4651_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_4652_high__def,axiom,
    ! [Xa: nat,Nb: nat] : vEBT_VEBT_high(Xa,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% high_def
tff(fact_4653_high__bound__aux,axiom,
    ! [Ma: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$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_4654_high__inv,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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))),Xa),Nb) = Y ) ) ).

% high_inv
tff(fact_4655_log__ceil__idem,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),Xa)) = archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,Xa)))) ) ) ).

% log_ceil_idem
tff(fact_4656_heigt__uplog__rel,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)) = archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_4657_delete__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_4658_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,Ta: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Deg) )
     => ( vEBT_invar_vebt(Ta,Deg)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_4659_log__one,axiom,
    ! [A2: real] : log(A2,one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_4660_delete__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_4661_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),log(A2,Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_4662_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),log(A2,Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_4663_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),log(A2,Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_4664_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),log(A2,Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_4665_log__less__cancel__iff,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),log(A2,Xa)),log(A2,Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_4666_log__eq__one,axiom,
    ! [A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(A2,A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_4667_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),log(A2,Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_4668_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(A2,Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_4669_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),log(A2,Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_4670_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(A2,Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_4671_log__le__cancel__iff,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(A2,Xa)),log(A2,Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_4672_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(A2,aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_4673_log__base__change,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(B2,Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),log(A2,Xa)),log(A2,B2)) ) ) ) ).

% log_base_change
tff(fact_4674_log__mult,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( log(A2,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),log(A2,Xa)),log(A2,Y)) ) ) ) ) ) ).

% log_mult
tff(fact_4675_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(B2,M)) ) ) ).

% le_log_of_power
tff(fact_4676_log__divide,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( log(A2,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,minus_minus(real,log(A2,Xa)),log(A2,Y)) ) ) ) ) ) ).

% log_divide
tff(fact_4677_log__base__pow,axiom,
    ! [A2: real,Nb: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),Nb),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),log(A2,Xa)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log_base_pow
tff(fact_4678_log__nat__power,axiom,
    ! [Xa: real,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( log(B2,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(B2,Xa)) ) ) ).

% log_nat_power
tff(fact_4679_log__of__power__less,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),log(B2,aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_less
tff(fact_4680_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
         => ( ( B2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
             => ( log(A2,Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),ln_ln(real,B2)),ln_ln(real,A2))),log(B2,Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_4681_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,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_4682_log__of__power__le,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(B2,aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_le
tff(fact_4683_less__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_4684_le__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_4685_log2__of__power__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_less
tff(fact_4686_pred__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_4687_succ__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_4688_log2__of__power__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_le
tff(fact_4689_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))),exp(real,one_one(real)))),ln_ln(real,Xa)) ) ) ).

% log_base_10_eq2
tff(fact_4690_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),ln_ln(real,exp(real,one_one(real)))),ln_ln(real,aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),ln_ln(real,Xa)) ) ) ).

% log_base_10_eq1
tff(fact_4691_pred__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_4692_succ__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_4693_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archimedean_ceiling(real,log(aa(nat,real,semiring_1_of_nat(real),B2),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_4694_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_4695_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,log(aa(nat,real,semiring_1_of_nat(real),B2),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_4696_htt__vebt__memberi__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_memberi(Tib,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_ag(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Ta),Tib),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),nat2(archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_4697_htt__vebt__inserti__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_inserti(Tib,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),nat2(archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_4698_nested__mint,axiom,
    ! [Mi: 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Nb)
     => ( ( Nb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Mi)
         => ( ( Ma != Mi )
           => aa(nat,$o,aa(nat,fun(nat,$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))),aa(nat,nat,aa(nat,fun(nat,nat),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,aa(nat,nat,aa(nat,fun(nat,nat),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_4699_summaxma,axiom,
    ! [Mi: 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi != Ma )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summary)) = vEBT_VEBT_high(Ma,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_4700_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_4701_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_4702_nat__0__iff,axiom,
    ! [I: int] :
      ( ( nat2(I) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_4703_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( nat2(Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_4704_nat__neg__numeral,axiom,
    ! [K: num] : nat2(uminus_uminus(int,aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_4705_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W)),nat2(Z))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% zless_nat_conj
tff(fact_4706_nat__zminus__int,axiom,
    ! [Nb: nat] : nat2(uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),Nb))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_4707_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_4708_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),nat2(Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_4709_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => ( aa(nat,A,semiring_1_of_nat(A),nat2(Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_4710_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_4711_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(archimedean_ceiling(real,Xa))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_4712_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),nat2(Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_4713_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Nb)),nat2(A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4714_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xa: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Nb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_4715_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Nb)),nat2(A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4716_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xa: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Nb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_4717_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_4718_option_Osel,axiom,
    ! [A: $tType,X2: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X2)) = X2 ).

% option.sel
tff(fact_4719_nat__zero__as__int,axiom,
    zero_zero(nat) = nat2(zero_zero(int)) ).

% nat_zero_as_int
tff(fact_4720_nat__mono,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(Xa)),nat2(Y)) ) ).

% nat_mono
tff(fact_4721_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z6)
       => ( ( nat2(Z) = nat2(Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_4722_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ! [X3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X3)
         => aa(nat,$o,P,nat2(X3)) ) ) ).

% all_nat
tff(fact_4723_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ? [X3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X3)
          & aa(nat,$o,P,nat2(X3)) ) ) ).

% ex_nat
tff(fact_4724_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_4725_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W)),nat2(Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_mono_iff
tff(fact_4726_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),nat2(Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_4727_nat__le__iff,axiom,
    ! [Xa: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(Xa)),Nb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_le_iff
tff(fact_4728_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),nat2(archimedean_ceiling(A,R2)))) ) ).

% of_nat_ceiling
tff(fact_4729_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) ) ).

% nat_0_le
tff(fact_4730_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = nat2(Z) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_4731_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,abs_abs(int),W))),nat2(aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_4732_Option_Othese__def,axiom,
    ! [A: $tType,A3: set(option(A))] : these(A,A3) = aa(set(option(A)),set(A),image(option(A),A,the2(A)),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_gc(set(option(A)),fun(option(A),$o),A3))) ).

% Option.these_def
tff(fact_4733_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),nat2(archimedean_ceiling(real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_4734_nat__plus__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X4),Xa2) = 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),Xa2))) ).

% nat_plus_as_int
tff(fact_4735_nat__times__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa2) = 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),Xa2))) ).

% nat_times_as_int
tff(fact_4736_nat__minus__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,minus_minus(nat,X4),Xa2) = nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_minus_as_int
tff(fact_4737_nat__div__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),Xa2) = nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_div_as_int
tff(fact_4738_nat__mod__as__int,axiom,
    ! [X4: nat,Xa2: nat] : modulo_modulo(nat,X4,Xa2) = nat2(modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X4),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_mod_as_int
tff(fact_4739_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W)),nat2(Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_4740_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = nat2(W) )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_4741_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( nat2(W) = M )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_4742_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,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_4743_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),W)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(W)),nat2(Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_4744_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z6)
       => ( nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z6)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(Z)),nat2(Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_4745_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),nat2(K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ) ).

% le_nat_iff
tff(fact_4746_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_4747_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z6)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(Z)),nat2(Z6)) ) ) ).

% nat_mult_distrib
tff(fact_4748_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z6)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z6),Z)
       => ( nat2(aa(int,int,minus_minus(int,Z),Z6)) = aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_4749_nat__diff__distrib_H,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( nat2(aa(int,int,minus_minus(int,Xa),Y)) = aa(nat,nat,minus_minus(nat,nat2(Xa)),nat2(Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_4750_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_4751_nat__div__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(Xa)),nat2(Y)) ) ) ).

% nat_div_distrib
tff(fact_4752_nat__div__distrib_H,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(Xa)),nat2(Y)) ) ) ).

% nat_div_distrib'
tff(fact_4753_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( nat2(aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),nat2(Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_4754_nat__mod__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( nat2(modulo_modulo(int,Xa,Y)) = modulo_modulo(nat,nat2(Xa),nat2(Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_4755_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
         => ( aa(int,word(A),ring_1_of_int(word(A)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),nat2(Xa)) ) ) ) ).

% word_of_int_nat
tff(fact_4756_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_4757_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,nat,suc,nat2(Z)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_4758_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(W)),M)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_4759_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z6)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(uminus_uminus(int,Z))),nat2(uminus_uminus(int,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_4760_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      nat2(aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2),aa(nat,nat,minus_minus(nat,B2),A2),aa(nat,nat,minus_minus(nat,A2),B2)) ).

% nat_abs_int_diff
tff(fact_4761_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_4762_diff__nat__eq__if,axiom,
    ! [Z: int,Z6: int] :
      aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z6)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z6),zero_zero(int)),
        nat2(Z),
        $let(
          d: int,
          d:= aa(int,int,minus_minus(int,Z),Z6),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),nat2(d)) ) ) ).

% diff_nat_eq_if
tff(fact_4763_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,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),uminus_uminus(A,aa(nat,A,semiring_1_of_nat(A),nat2(uminus_uminus(int,K)))),aa(nat,A,semiring_1_of_nat(A),nat2(K))) ) ).

% of_int_of_nat
tff(fact_4764_pred__less__length__list,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4765_pred__lesseq__max,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4766_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_4767_bit__split__inv,axiom,
    ! [Xa: nat,D: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xa,D),vEBT_VEBT_low(Xa,D),D) = Xa ).

% bit_split_inv
tff(fact_4768_low__def,axiom,
    ! [Xa: nat,Nb: nat] : vEBT_VEBT_low(Xa,Nb) = modulo_modulo(nat,Xa,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% low_def
tff(fact_4769_low__inv,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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))),Xa),Nb) = Xa ) ) ).

% low_inv
tff(fact_4770_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_4771_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),Xa)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_4772_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_4773_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_4774_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list(vEBT_VEBT),Xa: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mi),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_4775_insert__simp__norm,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_4776_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xa: 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,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ~ vEBT_VEBT_minNull(Newnode)
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4777_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xa: nat,Mi: 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)] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( ~ vEBT_VEBT_minNull(Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4778_del__x__not__mia,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
               => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4779_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xa: 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,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( vEBT_VEBT_minNull(Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4780_del__x__not__mi,axiom,
    ! [Mi: nat,Xa: 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,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                   => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $ite(
                          vEBT_VEBT_minNull(Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,H),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4781_del__x__mia,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(
                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_4782_del__x__mi__lets__in__minNull,axiom,
    ! [Xa: nat,Mi: 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] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4783_del__x__mi__lets__in,axiom,
    ! [Xa: nat,Mi: 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)] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $ite(
                            vEBT_VEBT_minNull(Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,H),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4784_del__x__mi,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_4785_del__in__range,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                xn: nat,
                xn:= 
                  $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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)))))),Xa),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xa = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                  $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_4786_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_4787_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(Xa,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_4788_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),Va2: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,Va2),TreeList,S2),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_4789_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [Va2: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,Va2),TreeList,Vd),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_4790_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(aa(nat,$o,aa(nat,fun(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(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4791_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,Va2),TreeList,Vc),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4792_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_4793_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_4794_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_4795_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2)
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4796_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Nb)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                      & aa(nat,$o,aa(nat,fun(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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_4797_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_4798_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xa != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2)
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4799_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4800_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xa = Mi,
          zero_zero(nat),
          $ite(
            Xa = Ma,
            zero_zero(nat),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
              zero_zero(nat),
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(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(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                  zero_zero(nat) ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_4801_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_i_n_s_e_r_t2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_4802_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Xa,Mi)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_4803_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Nb) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Nb)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                      & aa(nat,$o,aa(nat,fun(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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_4804_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_4805_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
           => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_4806_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi2,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma2,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa)
                                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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_4807_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] : Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] : Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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_4808_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: $o,B3: $o] : A1 = vEBT_Leaf((A4),(B3))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,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_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,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_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( aa(set(vEBT_VEBT),$o,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,$o,aa(nat,fun(nat,$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_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_12)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_1) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
                                   => ( aa(nat,$o,aa(nat,fun(nat,$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))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X4)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_invar_vebt(X4,N) )
                       => ( vEBT_invar_vebt(Summary2,M2)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                           => ( ( M2 = aa(nat,nat,suc,N) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                               => ( ! [I4: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                     => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_12)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                         => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_1) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X4)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_4809_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList4: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList4,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X3,N6) )
            & vEBT_invar_vebt(Summary3,N6)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = 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_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X3: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList4: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList4,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X3,N6) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N6))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = 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_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X3: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList4: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList4,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X3,N6) )
            & vEBT_invar_vebt(Summary3,N6)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = 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) )
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6))
                 => ( ( ( vEBT_VEBT_high(Ma3,N6) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(Ma3,N6)) )
                    & ! [X3: nat] :
                        ( ( ( vEBT_VEBT_high(X3,N6) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(X3,N6)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X3)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma3) ) ) ) ) ) )
        | ? [TreeList4: list(vEBT_VEBT),N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList4,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X3,N6) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N6))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),aa(nat,nat,suc,N6)) )
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N6) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(Ma3,N6)) )
                    & ! [X3: nat] :
                        ( ( ( vEBT_VEBT_high(X3,N6) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(X3,N6)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X3)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_4810_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B3))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts,S) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ) )
           => ( ! [V2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),
                                    aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Xaa,Mi2)),
                                    aa(nat,nat,aa(nat,fun(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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_4811_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(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_4812_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(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_4813_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
        one_one(nat),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          one_one(nat),
          $let(
            xn: nat,
            xn:= 
              $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xa),
            $let(
              l: nat,
              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(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_4814_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
        aa(nat,option(nat),some(nat),Mi),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summary,h),
                    $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_4815_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
        aa(nat,option(nat),some(nat),Ma),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summary,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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_4816_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
          $let(
            xn: nat,
            xn:= 
              $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xa),
            $let(
              minn: nat,
              minn:= 
                $ite(Xa = Mi,xn,Mi),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summary,h),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Ma,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                  Ma ))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,sn) ),
                        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_4817_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xa = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = 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] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(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_4818_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xa = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(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] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(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_4819_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(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_4820_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(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_4821_vebt__delete_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B3)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != vEBT_Leaf($false,(B3)) ) ) )
       => ( ! [A4: $o] :
              ( ? [B3: $o] : Xa = vEBT_Leaf((A4),(B3))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf((A4),$false) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B3)) )
               => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
                 => ( Y != vEBT_Leaf((A4),(B3)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                   => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                     => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( 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 = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $let(
                                                          maxs: option(nat),
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                    $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_4822_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(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] : Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                              one_one(nat),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                one_one(nat),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(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_4823_vebt__succ_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( Xa = 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] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            aa(nat,option(nat),some(nat),Mi2),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_4824_vebt__pred_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] : Xa = vEBT_Leaf((A4),(Uw2))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A4),(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((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2)
                 => ( Y != none(nat) ) )
               => ( ( ? [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              aa(nat,option(nat),some(nat),Ma2),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa),aa(nat,option(nat),some(nat),Mi2),none(nat)),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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_4825_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xa = 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] : Xa = 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] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(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_4826_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xa = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A4),(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] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(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_4827_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = 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,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
          | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
          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(
              ( ( Xa = Mi )
              & ( Xa = 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(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                    one_one(nat))),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xa),
                  $let(
                    l: nat,
                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $let(
                      h: nat,
                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(
                        aa(nat,$o,aa(nat,fun(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_4828_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(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] : Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                    aa(nat,nat,
                                      aa(nat,fun(nat,nat),plus_plus(nat),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                              $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(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 = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(
                                              aa(nat,$o,aa(nat,fun(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_4829_in__children__def,axiom,
    ! [Nb: nat,TreeList: list(vEBT_VEBT),Xa: nat] :
      ( vEBT_V5917875025757280293ildren(Nb,TreeList,Xa)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,Nb))),vEBT_VEBT_low(Xa,Nb)) ) ).

% in_children_def
tff(fact_4830_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                    | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                      $ite(
                                        ( ( Xaa = Mi2 )
                                        & ( Xaa = Ma2 ) ),
                                        aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,
                                              aa(nat,fun(nat,nat),plus_plus(nat),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                                  $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(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 = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(
                                                  aa(nat,$o,aa(nat,fun(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))),
                                                              $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                                  one_one(nat) ) ) ) )) )) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4831_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(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))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4832_vebt__succ_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = none(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                aa(nat,option(nat),some(nat),Mi2),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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),
                                            $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_4833_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(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)),
                              $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                    one_one(nat),
                                    $let(
                                      l: nat,
                                      l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $let(
                                        h: nat,
                                        h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(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))),
                                                      $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                            one_one(nat) )) ) ) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4834_vebt__pred_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(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)),
                              $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = none(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = none(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  aa(nat,option(nat),some(nat),Ma2),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa),aa(nat,option(nat),some(nat),Mi2),none(nat)),
                                                aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                        none(nat) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_4835_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                  one_one(nat),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    one_one(nat),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(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)),
                                              $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                                            one_one(nat) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4836_vebt__delete_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf($false,(B3)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf((A4),$false) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = vEBT_Leaf((A4),(B3)) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( 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 = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $let(
                                                              maxs: option(nat),
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                        $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_4837_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(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)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4838_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B3)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(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)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
                                        one_one(nat) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4839_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B3))) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S)),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                      aa(nat,product_prod(nat,nat),
                                        aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Xaa,Mi2)),
                                        aa(nat,nat,aa(nat,fun(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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4840_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S)),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4841_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi2,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa)
                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                        zero_zero(nat) ) ) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4842_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
         => ( ( ? [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,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 = Mi2,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Ma2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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_4843_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xa = vEBT_Leaf((A4),(B3))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] : Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] : Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                            $let(
                              h: nat,
                              h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                & ~ ( ( Xaa = Mi2 )
                                    | ( Xaa = Ma2 ) ) ),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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_4844_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)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts2,S2),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_4845_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)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_4846_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,Xa: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A2),(B2)),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(Xa = 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_4847_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,Xa: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xa) = 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_4848_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,
    ! [Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = 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_4849_insersimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_4850_insertsimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_4851_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,
    ! [Va2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xa: 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)),Va2),zero_zero(nat),Uy,Uz),Xa) = 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_4852_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,
    ! [Va2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: 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)),Va2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xa) = 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_4853_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,Xa: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A2),(B2)),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(Xa = 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_4854_insert__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))) ) ).

% insert_bound_height
tff(fact_4855_member__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r(Ta,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_4856_insert__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_4857_member__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(Ta,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% member_bound_size_univ
tff(fact_4858_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(
          Xa = Mi,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xa = Ma,
              one_one(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                      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(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(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(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_4859_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = 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,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
              & ~ ( ( Xa = Mi )
                  | ( Xa = Ma ) ) ),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_4860_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S)),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,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,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                    & ~ ( ( Xaa = Mi2 )
                                        | ( Xaa = Ma2 ) ) ),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4861_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,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,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 = Mi2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xaa = Ma2,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                            one_one(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),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_4862_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4863_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4864_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4865_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList2: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4866_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList2: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4867_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList2: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList2,S)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4868_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2)),Xaa)) ) )
               => ~ ! [V2: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4869_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2) )
                   => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2)),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4870_lowi__h,axiom,
    ! [Xa: nat,Nb: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_lowi(Xa,Nb),aa(nat,fun(nat,assn),aTP_Lamp_pz(nat,fun(nat,fun(nat,assn)),Xa),Nb)) ).

% lowi_h
tff(fact_4871_TBOUND__lowi,axiom,
    ! [Xa: nat,Nb: nat] : time_TBOUND(nat,vEBT_VEBT_lowi(Xa,Nb),one_one(nat)) ).

% TBOUND_lowi
tff(fact_4872_lowi__hT,axiom,
    ! [Xa: nat,Nb: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_lowi(Xa,Nb),aa(nat,fun(nat,assn),aTP_Lamp_pz(nat,fun(nat,fun(nat,assn)),Xa),Nb),one_one(nat)) ).

% lowi_hT
tff(fact_4873_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList2,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList2,Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4874_highi__h,axiom,
    ! [Xa: nat,Nb: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_highi(Xa,Nb),aa(nat,fun(nat,assn),aTP_Lamp_qa(nat,fun(nat,fun(nat,assn)),Xa),Nb)) ).

% highi_h
tff(fact_4875_highi__hT,axiom,
    ! [Xa: nat,Nb: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_highi(Xa,Nb),aa(nat,fun(nat,assn),aTP_Lamp_qa(nat,fun(nat,fun(nat,assn)),Xa),Nb),one_one(nat)) ).

% highi_hT
tff(fact_4876_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_qb(real,fun(nat,real),Xa)) ) ).

% monoseq_arctan_series
tff(fact_4877_TBOUND__highi,axiom,
    ! [Xa: nat,Nb: nat] : time_TBOUND(nat,vEBT_VEBT_highi(Xa,Nb),one_one(nat)) ).

% TBOUND_highi
tff(fact_4878_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),Xa)) ) ) ).

% monoseq_realpow
tff(fact_4879_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_4880_set__encode__insert,axiom,
    ! [A3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( ~ aa(set(nat),$o,member(nat,Nb),A3)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,Nb),A3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_4881_butlast__upd__last__eq,axiom,
    ! [A: $tType,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),L))
     => ( list_update(A,butlast(A,L),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa) = append(A,take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),bit0(one2))),L),aa(list(A),list(A),cons(A,Xa),nil(A))) ) ) ).

% butlast_upd_last_eq
tff(fact_4882_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se5668285175392031749et_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_4883_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se5668285175392031749et_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_4884_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_4885_length__butlast,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),butlast(A,Xs)) = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_butlast
tff(fact_4886_take__butlast__conv,axiom,
    ! [A: $tType,L: list(A)] : take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,L) ).

% take_butlast_conv
tff(fact_4887_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),bit_se5668285175392031749et_bit(int,Nb,K)) ).

% set_bit_greater_eq
tff(fact_4888_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_4889_butlast__subset,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( ( Xs != nil(A) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),butlast(A,Xs))),A3) ) ) ).

% butlast_subset
tff(fact_4890_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),butlast(A,Xs)) ) ) ) ).

% sorted_butlast
tff(fact_4891_nth__butlast,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),butlast(A,Xs)))
     => ( aa(nat,A,nth(A,butlast(A,Xs)),Nb) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% nth_butlast
tff(fact_4892_take__butlast,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,Nb,butlast(A,Xs)) = take(A,Nb,Xs) ) ) ).

% take_butlast
tff(fact_4893_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_4894_takeWhile__not__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( takeWhile(A,aTP_Lamp_qc(list(A),fun(A,$o),Xs),Xs) = butlast(A,Xs) ) ) ).

% takeWhile_not_last
tff(fact_4895_butlast__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xa: A] :
      butlast(A,list_update(A,Xs,K,Xa)) = $ite(K = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),butlast(A,Xs),list_update(A,butlast(A,Xs),K,Xa)) ).

% butlast_list_update
tff(fact_4896_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : butlast(A,Xs) = take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ).

% butlast_conv_take
tff(fact_4897_hd__butlast,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),A,hd(A),butlast(A,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_butlast
tff(fact_4898_butlast__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( butlast(A,take(A,Nb,Xs)) = take(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_4899_take__minus__one__conv__butlast,axiom,
    ! [A: $tType,Nb: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),L))
     => ( take(A,aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,take(A,Nb,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_4900_set__encode__def,axiom,
    nat_set_encode = aa(fun(nat,nat),fun(set(nat),nat),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_4901_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_4902_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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),uminus_uminus(real,one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),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_4903_ln__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( ln_ln(real,Xa) = suminf(real,aTP_Lamp_qd(real,fun(nat,real),Xa)) ) ) ) ).

% ln_series
tff(fact_4904_arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( arctan(Xa) = suminf(real,aTP_Lamp_qe(real,fun(nat,real),Xa)) ) ) ).

% arctan_series
tff(fact_4905_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_4906_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_4907_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_4908_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_4909_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,A2),C2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2))
            <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_4910_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,Xa),uminus_uminus(A,Y))
        <=> aa(A,$o,dvd_dvd(A,Xa),Y) ) ) ).

% dvd_minus_iff
tff(fact_4911_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,uminus_uminus(A,Xa)),Y)
        <=> aa(A,$o,dvd_dvd(A,Xa),Y) ) ) ).

% minus_dvd_iff
tff(fact_4912_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,M),aa(A,A,abs_abs(A),K))
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% dvd_abs_iff
tff(fact_4913_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,abs_abs(A),M)),K)
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% abs_dvd_iff
tff(fact_4914_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_4915_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_4916_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_4917_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_4918_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_4919_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_4920_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_4921_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_4922_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_4923_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_4924_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_4925_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_4926_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_4927_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_4928_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_4929_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_4930_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_4931_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_4932_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_4933_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_qf(nat,fun(nat,$o),Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_4934_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_4935_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ) ) ).

% unit_mult_div_div
tff(fact_4936_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(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_4937_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_4938_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_4939_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_4940_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F: fun(nat,A)] : suminf(A,aTP_Lamp_qg(fun(nat,A),fun(nat,A),F)) = aa(nat,A,F,zero_zero(nat)) ) ).

% powser_zero
tff(fact_4941_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_4942_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_4943_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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_4944_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4945_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(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,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_4946_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(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,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4947_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_4948_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,aa(nat,fun(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_4949_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_4950_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_4951_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4952_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,minus_minus(nat,Nb),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_4953_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W))
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4954_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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_4955_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,$o,aa(nat,fun(nat,$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_4956_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_4957_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_4958_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_4959_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_qh(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_qh(A,fun(A,$o),B2)))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_4960_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,dvd_dvd(A,A2),A2) ) ).

% dvd_refl
tff(fact_4961_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_4962_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_4963_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => aa(A,$o,dvd_dvd(A,L),K) ) ) ).

% dvd_if_abs_eq
tff(fact_4964_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)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_4965_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( aa(A,$o,dvd_dvd(A,C2),A2)
           => ( aa(A,$o,dvd_dvd(A,C2),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4966_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4967_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_4968_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_4969_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_4970_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_4971_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_4972_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_4973_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_4974_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_4975_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_4976_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_4977_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_4978_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_4979_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,Y3: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3) )
               => ( aa(A,$o,dvd_dvd(A,X),A2)
                 => ~ aa(A,$o,dvd_dvd(A,Y3),B2) ) ) ) ) ).

% dvd_productE
tff(fact_4980_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))
         => ? [B11: A,C9: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B11),C9) )
              & aa(A,$o,dvd_dvd(A,B11),B2)
              & aa(A,$o,dvd_dvd(A,C9),C2) ) ) ) ).

% division_decomp
tff(fact_4981_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,dvd_dvd(A,Xa),Y)
         => ( aa(A,$o,dvd_dvd(A,Xa),Z)
           => aa(A,$o,dvd_dvd(A,Xa),aa(A,A,minus_minus(A,Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_4982_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_4983_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_4984_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_4985_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,dvd_dvd(nat,A2),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_4986_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_4987_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_4988_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_4989_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_4990_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_qh(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_qh(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_4991_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_4992_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_4993_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_4994_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_4995_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% minf(10)
tff(fact_4996_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% minf(9)
tff(fact_4997_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% pinf(10)
tff(fact_4998_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S2: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2))
          <=> aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S2)) ) ) ) ).

% pinf(9)
tff(fact_4999_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5000_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_5001_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_5002_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_5003_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_5004_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_5005_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_5006_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_5007_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_5008_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_5009_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_5010_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_5011_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_5012_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_5013_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_5014_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)
         => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_5015_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5016_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),uminus_uminus(A,A2)),B2) = uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_5017_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),uminus_uminus(A,B2)) = uminus_uminus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_5018_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_5019_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_5020_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_5021_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: A,Y: A,Nb: nat,M: nat] :
          ( aa(A,$o,dvd_dvd(A,Xa),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_5022_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat,B2: A,M: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2) ) ) ) ).

% power_le_dvd
tff(fact_5023_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% le_imp_power_dvd
tff(fact_5024_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_5025_dvd__pos__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,dvd_dvd(nat,M),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_5026_nat__dvd__not__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
       => ~ aa(nat,$o,dvd_dvd(nat,Nb),M) ) ) ).

% nat_dvd_not_less
tff(fact_5027_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,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
        | aa(nat,$o,dvd_dvd(nat,M),Nb) ) ) ).

% dvd_minus_self
tff(fact_5028_dvd__diffD,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Nb))
     => ( aa(nat,$o,dvd_dvd(nat,K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => aa(nat,$o,dvd_dvd(nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_5029_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Nb))
     => ( aa(nat,$o,dvd_dvd(nat,K),M)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => aa(nat,$o,dvd_dvd(nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_5030_less__eq__dvd__minus,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,dvd_dvd(nat,M),Nb)
      <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_5031_bezout__lemma__nat,axiom,
    ! [D: nat,A2: nat,B2: nat,Xa: 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),Xa) = 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),Xa) = 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,Y3: 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)),Y3)),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),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_5032_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X: nat,Y3: 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),Y3)),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),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_5033_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X: nat,Y3: 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),Y3)) = 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),Y3)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_5034_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => aa(A,$o,dvd_dvd(A,semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,M)) ) ) ).

% fact_dvd
tff(fact_5035_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X3: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X3))
        <=> ? [X3: A] :
              ( aa(A,$o,dvd_dvd(A,L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),zero_zero(A)))
              & aa(A,$o,P,X3) ) ) ) ).

% unity_coeff_ex
tff(fact_5036_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) )
             => ! [C5: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C5) ) ) ) ).

% unit_dvdE
tff(fact_5037_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5038_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,A2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5039_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
            <=> aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5040_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
            <=> aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5041_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)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),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_5042_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D: A,D4: A,Ta: A] :
          ( aa(A,$o,dvd_dvd(A,D),D4)
         => ! [X4: A,K5: A] :
              ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Ta))
            <=> ~ aa(A,$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),D4))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_5043_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D: A,D4: A,Ta: A] :
          ( aa(A,$o,dvd_dvd(A,D),D4)
         => ! [X4: A,K5: A] :
              ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Ta))
            <=> aa(A,$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),D4))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_5044_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5045_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_5046_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_5047_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_5048_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_5049_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = C2 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_5050_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_5051_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_5052_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% dvd_imp_le
tff(fact_5053_prod__dvd__prod__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B4: set(A),A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => aa(B,$o,dvd_dvd(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B4)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_5054_prod__dvd__prod__subset2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [B4: set(A),A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(B,$o,dvd_dvd(B,aa(A,B,F,A4)),aa(A,B,G,A4)) )
             => aa(B,$o,dvd_dvd(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_5055_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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,dvd_dvd(nat,M),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_5056_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_5057_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_5058_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D2: nat,X: nat,Y3: 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),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_5059_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_5060_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,M: nat,Q4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( ( modulo_modulo(nat,M,Q4) = modulo_modulo(nat,Nb,Q4) )
      <=> aa(nat,$o,dvd_dvd(nat,Q4),aa(nat,nat,minus_minus(nat,M),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5061_dvd__fact,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => aa(nat,$o,dvd_dvd(nat,M),semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_5062_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_qf(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_5063_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_5064_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_5065_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = B3 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B3) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5066_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5067_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5068_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xa: A,M: nat,Nb: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb))
          <=> ( aa(A,$o,dvd_dvd(A,Xa),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_5069_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat,Xa: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            | ( Xa = one_one(A) ) )
         => aa(A,$o,dvd_dvd(A,Xa),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Nb)) ) ) ).

% dvd_power
tff(fact_5070_div2__even__ext__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa)
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Y) )
       => ( Xa = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_5071_dvd__mult__cancel1,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_5072_dvd__mult__cancel2,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_5073_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Nb),K)))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_5074_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,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_5075_dvd__minus__add,axiom,
    ! [Q4: nat,Nb: nat,R2: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q4),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M))
       => ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Nb),Q4))
        <=> aa(nat,$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)),Q4))) ) ) ) ).

% dvd_minus_add
tff(fact_5076_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$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_5077_diff__mod__le,axiom,
    ! [A2: nat,D: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),D)
     => ( aa(nat,$o,dvd_dvd(nat,B2),D)
       => aa(nat,$o,aa(nat,fun(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_5078_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_5079_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_5080_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_odd
tff(fact_5081_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% odd_pos
tff(fact_5082_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,M,A2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_5083_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_5084_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A3: set(A),F: fun(A,B),B2: B] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(B,fun(A,B),aTP_Lamp_qi(fun(A,B),fun(B,fun(A,B)),F),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_qj(fun(A,B),fun(B,fun(A,$o)),F),B2))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_qk(fun(A,B),fun(B,fun(A,$o)),F),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_5085_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_5086_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_5087_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_5088_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_5089_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_5090_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_le_even_power
tff(fact_5091_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_even
tff(fact_5092_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_5093_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))
      <=> ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ) ) ).

% even_set_encode_iff
tff(fact_5094_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_5095_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_5096_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5097_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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_5098_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_5099_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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_5100_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xa))),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)),Xa)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_5101_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_5102_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
              & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5103_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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_5104_sum__split__even__odd,axiom,
    ! [F: fun(nat,real),G: fun(nat,real),Nb: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_ql(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F),G)),aa(nat,set(nat),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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qm(fun(nat,real),fun(nat,real),F)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qn(fun(nat,real),fun(nat,real),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ).

% sum_split_even_odd
tff(fact_5105_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = 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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_5106_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_5107_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = 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,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_5108_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_5109_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_5110_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),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),uminus_uminus(real,one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),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_5111_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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_5112_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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_5113_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = 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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_5114_VEBT__internal_OTb_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = 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(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_5115_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,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = 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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_5116_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,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = 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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_5117_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != one_one(int) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != one_one(int) ) )
         => ~ ! [N: nat] :
                ( ( Xa = 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(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_5118_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_qo(nat,fun(nat,A),Nb)),aa(nat,set(nat),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_5119_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_qp(nat,fun(nat,A),Nb)),aa(nat,set(nat),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_5120_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = one_one(int) )
             => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = one_one(int) )
               => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = 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(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_5121_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,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = 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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ aa(nat,$o,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_5122_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,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = 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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ aa(nat,$o,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_5123_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_qq(int,fun(int,$o),Nb))) = aa(int,int,abs_abs(int),Nb) ) ) ).

% Max_divisors_self_int
tff(fact_5124_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_5125_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_5126_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_5127_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_5128_udvd__imp__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] :
          ( udvd(A,Va2,W)
         => aa(word(A),$o,dvd_dvd(word(A),Va2),W) ) ) ).

% udvd_imp_dvd
tff(fact_5129_zdvd__antisym__nonneg,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
       => ( aa(int,$o,dvd_dvd(int,M),Nb)
         => ( aa(int,$o,dvd_dvd(int,Nb),M)
           => ( M = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_5130_zdvd__not__zless,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),Nb)
       => ~ aa(int,$o,dvd_dvd(int,Nb),M) ) ) ).

% zdvd_not_zless
tff(fact_5131_zdvd__mono,axiom,
    ! [K: int,M: int,Ta: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,M),Ta)
      <=> aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ta)) ) ) ).

% zdvd_mono
tff(fact_5132_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_5133_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_qq(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_5134_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa(int,$o,dvd_dvd(int,Z),Nb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_5135_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,D),I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_5136_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_5137_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_5138_zdvd__mult__cancel1,axiom,
    ! [M: int,Nb: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb)),M)
      <=> ( aa(int,int,abs_abs(int),Nb) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_5139_int__div__sub__1,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),M)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,Nb),one_one(int))),M) = $ite(aa(int,$o,dvd_dvd(int,M),Nb),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),M)),one_one(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),M)) ) ) ).

% int_div_sub_1
tff(fact_5140_exp__dvd__iff__exp__udvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(word(A),$o,dvd_dvd(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)),W)
        <=> udvd(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),W) ) ) ).

% exp_dvd_iff_exp_udvd
tff(fact_5141_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,nat2(Z)),M)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),aa(int,$o,dvd_dvd(int,Z),aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_5142_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),nat2(K))
      <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_5143_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,$o,aa(int,fun(int,$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_5144_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,$o,aa(int,fun(int,$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_5145_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = 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:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5146_VEBT__internal_OTb_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
             => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
               => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = 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(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_5147_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = 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,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_5148_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = 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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_5149_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_5150_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_qr(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_5151_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N7: set(nat),F: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N7)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),N7)
               => ( aa(nat,A,F,N) = zero_zero(A) ) )
           => ( suminf(A,F) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),N7) ) ) ) ) ).

% suminf_finite
tff(fact_5152_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_5153_pi__series,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_qs(nat,real)) ).

% pi_series
tff(fact_5154_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,Xa) = Y ) ) ) ) ).

% round_unique
tff(fact_5155_round__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int] : archimedean_round(A,aa(int,A,ring_1_of_int(A),Nb)) = Nb ) ).

% round_of_int
tff(fact_5156_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_5157_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_5158_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_5159_round__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: nat] : archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ).

% round_of_nat
tff(fact_5160_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: num] : archimedean_round(A,uminus_uminus(A,aa(num,A,numeral_numeral(A),Nb))) = uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb)) ) ).

% round_neg_numeral
tff(fact_5161_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_5162_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_5163_pi__not__less__zero,axiom,
    ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_5164_pi__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_5165_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_5166_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% ceiling_ge_round
tff(fact_5167_pi__ge__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_5168_pi__half__neq__zero,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_5169_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_le_two
tff(fact_5170_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_5171_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_5172_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_ge_zero
tff(fact_5173_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_5174_arctan__inverse,axiom,
    ! [Xa: real] :
      ( ( Xa != zero_zero(real) )
     => ( arctan(aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xa)) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Xa)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),arctan(Xa)) ) ) ).

% arctan_inverse
tff(fact_5175_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_5176_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_ge
tff(fact_5177_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_gt
tff(fact_5178_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),Xa))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_5179_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xa),aa(int,A,ring_1_of_int(A),Nb)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,Xa) = Nb ) ) ) ).

% round_unique'
tff(fact_5180_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),T6)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),zero_zero(real))
            & ( cos(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qt(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_5181_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),Xa)
            & ( cos(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qt(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_5182_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),Xa)
            & ( sin(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qu(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_5183_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_5184_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_5185_sin__pi,axiom,
    sin(real,pi) = zero_zero(real) ).

% sin_pi
tff(fact_5186_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: 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,Xa)),cos(A,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xa)),sin(A,Xa))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_5187_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_5188_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_5189_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_5190_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_5191_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_5192_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_5193_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_5194_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_5195_cos__pi__eq__zero,axiom,
    ! [M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))))),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_5196_sin__cos__npi,axiom,
    ! [Nb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),uminus_uminus(real,one_one(real))),Nb) ).

% sin_cos_npi
tff(fact_5197_sincos__principal__value,axiom,
    ! [Xa: real] :
    ? [Y3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,pi)),Y3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),pi)
      & ( sin(real,Y3) = sin(real,Xa) )
      & ( cos(real,Y3) = cos(real,Xa) ) ) ).

% sincos_principal_value
tff(fact_5198_sin__zero__abs__cos__one,axiom,
    ! [Xa: real] :
      ( ( sin(real,Xa) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,Xa)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_5199_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) = one_one(A) )
         => ( sin(A,Xa) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_5200_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sin(A,Xa) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,Xa)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_5201_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : sin(A,aa(A,A,minus_minus(A,Xa),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xa)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),sin(A,Y))) ) ).

% sin_diff
tff(fact_5202_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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,Xa)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),sin(A,Y))) ) ).

% sin_add
tff(fact_5203_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xa)),sin(A,Y))) ) ).

% cos_add
tff(fact_5204_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : cos(A,aa(A,A,minus_minus(A,Xa),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,Xa)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xa)),sin(A,Y))) ) ).

% cos_diff
tff(fact_5205_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = 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,Xa))),cos(A,Xa)) ) ).

% sin_double
tff(fact_5206_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_5207_sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),one_one(real)) ).

% sin_le_one
tff(fact_5208_cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),one_one(real)) ).

% cos_le_one
tff(fact_5209_abs__sin__x__le__abs__x,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xa))),aa(real,real,abs_abs(real),Xa)) ).

% abs_sin_x_le_abs_x
tff(fact_5210_cos__arctan__not__zero,axiom,
    ! [Xa: real] : cos(real,arctan(Xa)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_5211_sin__cos__le1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,Xa)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,Xa)),cos(real,Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_5212_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero
tff(fact_5213_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,Xa)),sin(real,Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_5214_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_ge_zero
tff(fact_5215_sin__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),sin(real,Xa)) ).

% sin_ge_minus_one
tff(fact_5216_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_5217_cos__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_5218_cos__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( ( cos(real,Xa) = cos(real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_5219_cos__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),cos(real,Xa)) ).

% cos_ge_minus_one
tff(fact_5220_abs__sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xa))),one_one(real)) ).

% abs_sin_le_one
tff(fact_5221_abs__cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,Xa))),one_one(real)) ).

% abs_cos_le_one
tff(fact_5222_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Z),W)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_5223_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_5224_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_5225_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_5226_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_5227_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_sin
tff(fact_5228_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_5229_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_5230_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_5231_cos__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xa)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_5232_cos__monotone__0__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xa)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_5233_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,pi)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => ( ( sin(real,Xa) = zero_zero(real) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_5234_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),pi)
     => ( ( sin(real,Xa) = zero_zero(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_5235_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Y)),cos(real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_5236_sin__zero__iff__int2,axiom,
    ! [Xa: real] :
      ( ( sin(real,Xa) = zero_zero(real) )
    <=> ? [I3: int] : Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),pi) ) ).

% sin_zero_iff_int2
tff(fact_5237_sincos__total__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),pi)
            & ( Xa = cos(real,T6) )
            & ( Y = sin(real,T6) ) ) ) ) ).

% sincos_total_pi
tff(fact_5238_sin__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% sin_expansion_lemma
tff(fact_5239_cos__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = uminus_uminus(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% cos_expansion_lemma
tff(fact_5240_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_5241_cos__two__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_5242_cos__is__zero,axiom,
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X) = zero_zero(real) )
      & ! [Y4: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y4) = zero_zero(real) ) )
         => ( Y4 = X ) ) ) ).

% cos_is_zero
tff(fact_5243_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_5244_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Y)),cos(real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_5245_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
            & ( cos(real,X) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),pi)
                  & ( cos(real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% cos_total
tff(fact_5246_sincos__total__pi__half,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xa = cos(real,T6) )
              & ( Y = sin(real,T6) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_5247_sincos__total__2pi__le,axiom,
    ! [Xa: 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),Xa),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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
          & aa(real,$o,aa(real,fun(real,$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))
          & ( Xa = cos(real,T6) )
          & ( Y = sin(real,T6) ) ) ) ).

% sincos_total_2pi_le
tff(fact_5248_sincos__total__2pi,axiom,
    ! [Xa: 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),Xa),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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
           => ( aa(real,$o,aa(real,fun(real,$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))
             => ( ( Xa = cos(real,T6) )
               => ( Y != sin(real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_5249_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_5250_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_5251_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_cos
tff(fact_5252_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero2
tff(fact_5253_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_5254_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$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))),Xa))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_5255_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_gt_zero
tff(fact_5256_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_5257_sin__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_5258_sin__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,Xa) = sin(real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_5259_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_5260_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: 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))),Xa)) = 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,Xa)),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,Xa))) ) ).

% cos_treble_cos
tff(fact_5261_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_5262_sin__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),uminus_uminus(real,pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_5263_sin__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_5264_sin__monotone__2pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Y)),sin(real,Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_5265_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% sin_total
tff(fact_5266_cos__gt__zero__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_5267_cos__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_ge_zero
tff(fact_5268_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_5269_sin__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( sin(real,Xa) = zero_zero(real) )
    <=> ? [I3: int] :
          ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I3)
          & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_5270_cos__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( cos(real,Xa) = zero_zero(real) )
    <=> ? [I3: int] :
          ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I3)
          & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_5271_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( sin(real,Xa) = zero_zero(real) )
       => ? [N: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_5272_sin__zero__iff,axiom,
    ! [Xa: real] :
      ( ( sin(real,Xa) = zero_zero(real) )
    <=> ( ? [N6: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N6: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xa = uminus_uminus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_5273_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( cos(real,Xa) = zero_zero(real) )
       => ? [N: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_5274_cos__zero__iff,axiom,
    ! [Xa: real] :
      ( ( cos(real,Xa) = zero_zero(real) )
    <=> ( ? [N6: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N6: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xa = uminus_uminus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_5275_Maclaurin__sin__expansion,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T6: real] : sin(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qu(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_5276_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T6: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xa))
      & ( sin(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qu(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_5277_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T6: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xa))
      & ( cos(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qt(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_5278_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ? [T6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),Xa)
          & ( sin(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qu(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_5279_lowi__def,axiom,
    ! [Xa: nat,Nb: nat] : vEBT_VEBT_lowi(Xa,Nb) = heap_Time_return(nat,modulo_modulo(nat,Xa,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% lowi_def
tff(fact_5280_highi__def,axiom,
    ! [Xa: nat,Nb: nat] : vEBT_VEBT_highi(Xa,Nb) = heap_Time_return(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% highi_def
tff(fact_5281_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,aa(num,fun(num,$o),ord_less(num),M),Nb),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),unique1321980374590559556d_step(A,aa(num,num,bit1,Nb),unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(aa(num,num,bit1,Nb))))) ) ).

% divmod_algorithm_code(8)
tff(fact_5282_highsimp,axiom,
    ! [Xa: nat,Nb: nat] : heap_Time_return(nat,vEBT_VEBT_high(Xa,Nb)) = vEBT_VEBT_highi(Xa,Nb) ).

% highsimp
tff(fact_5283_lowsimp,axiom,
    ! [Xa: nat,Nb: nat] : heap_Time_return(nat,vEBT_VEBT_low(Xa,Nb)) = vEBT_VEBT_lowi(Xa,Nb) ).

% lowsimp
tff(fact_5284_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_5285_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,bit0(Nb)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_5286_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,Nb)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_5287_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] :
          unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))),unique1321980374590559556d_step(A,aa(num,num,bit1,Nb),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,Nb))))) ) ).

% divmod_algorithm_code(7)
tff(fact_5288_return__wp__rule,axiom,
    ! [A: $tType,Q: fun(A,assn),Xa: A] : hoare_hoare_triple(A,aa(A,assn,Q,Xa),heap_Time_return(A,Xa),Q) ).

% return_wp_rule
tff(fact_5289_return__cons__rule,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xa: A] :
      ( entails(P,aa(A,assn,Q,Xa))
     => hoare_hoare_triple(A,P,heap_Time_return(A,Xa),Q) ) ).

% return_cons_rule
tff(fact_5290_return__sp__rule,axiom,
    ! [A: $tType,P: assn,Xa: A] : hoare_hoare_triple(A,P,heap_Time_return(A,Xa),aa(A,fun(A,assn),aTP_Lamp_qv(assn,fun(A,fun(A,assn)),P),Xa)) ).

% return_sp_rule
tff(fact_5291_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] :
          unique8689654367752047608divmod(A,M,Nb) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Nb),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,Nb,unique8689654367752047608divmod(A,M,bit0(Nb)))) ) ).

% divmod_divmod_step
tff(fact_5292_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),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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_5293_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => summable(real,aTP_Lamp_qe(real,fun(nat,real),Xa)) ) ).

% summable_arctan_series
tff(fact_5294_quickcheck__narrowing__samples_Onarrowing__samples_Opinduct,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),A0: code_integer,P: fun(code_integer,$o)] :
          ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),A0)
         => ( ! [I2: code_integer] :
                ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),I2)
               => ( ! [X4: product_prod(A,A),Xa2: A,Y4: A] :
                      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),I2)
                     => ( ( X4 = aa(code_integer,product_prod(A,A),A_of_integer,I2) )
                       => ( ( aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),Y4) = X4 )
                         => aa(code_integer,$o,P,aa(code_integer,code_integer,minus_minus(code_integer,I2),one_one(code_integer))) ) ) )
                 => aa(code_integer,$o,P,I2) ) )
           => aa(code_integer,$o,P,A0) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pinduct
tff(fact_5295_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_5296_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qw(nat,fun(fun(nat,A),fun(nat,A)),I),F)) ) ).

% summable_single
tff(fact_5297_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_qx(nat,A)) ) ).

% summable_zero
tff(fact_5298_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),K))
        <=> summable(A,F) ) ) ).

% summable_iff_shift
tff(fact_5299_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_5300_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qz(A,fun(fun(nat,A),fun(nat,A)),C2),F))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F) ) ) ) ).

% summable_cmult_iff
tff(fact_5301_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ra(fun(nat,A),fun(A,fun(nat,A)),F),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F) ) ) ) ).

% summable_divide_iff
tff(fact_5302_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rb(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F)) ) ) ).

% summable_If_finite
tff(fact_5303_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A3)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rc(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F)) ) ) ).

% summable_If_finite_set
tff(fact_5304_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_5305_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,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_5306_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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),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_5307_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A2) = uminus_uminus(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_5308_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_5309_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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,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_5310_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ix(fun(nat,A),fun(A,fun(nat,A)),F),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_rd(fun(nat,A),fun(A,fun(nat,real)),F),Z)) ) ) ) ).

% powser_insidea
tff(fact_5311_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N)),aa(nat,A,G,N))
         => ( summable(A,F)
           => ( summable(A,G)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_5312_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,F)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_5313_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_re(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ).

% summable_add
tff(fact_5314_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N7: nat,F: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,N))),aa(nat,real,G,N)) )
           => summable(A,F) ) ) ) ).

% summable_comparison_test'
tff(fact_5315_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A),G: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(A,F) ) ) ) ).

% summable_comparison_test
tff(fact_5316_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => summable(A,aTP_Lamp_rf(fun(nat,A),fun(nat,A),F)) ) ) ).

% summable_minus
tff(fact_5317_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,aTP_Lamp_rf(fun(nat,A),fun(nat,A),F))
        <=> summable(A,F) ) ) ).

% summable_minus_iff
tff(fact_5318_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,aTP_Lamp_rg(fun(nat,A),fun(nat,A),F))
        <=> summable(A,F) ) ) ).

% summable_Suc_iff
tff(fact_5319_summable__rabs__cancel,axiom,
    ! [F: fun(nat,real)] :
      ( summable(real,aTP_Lamp_rh(fun(nat,real),fun(nat,real),F))
     => summable(real,F) ) ).

% summable_rabs_cancel
tff(fact_5320_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ra(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_divide
tff(fact_5321_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ri(fun(nat,A),fun(nat,real),F))
         => summable(A,F) ) ) ).

% summable_norm_cancel
tff(fact_5322_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_rj(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_mult2
tff(fact_5323_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_rk(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_mult
tff(fact_5324_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => summable(B,aa(A,fun(nat,B),F,I2)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_rm(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F)) ) ) ).

% summable_sum
tff(fact_5325_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ).

% summable_diff
tff(fact_5326_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_ro(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_5327_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N7: set(nat),F: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N7)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),N7)
               => ( aa(nat,A,F,N) = zero_zero(A) ) )
           => summable(A,F) ) ) ) ).

% summable_finite
tff(fact_5328_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qz(A,fun(fun(nat,A),fun(nat,A)),C2),F))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F) ) ) ) ).

% summable_mult_D
tff(fact_5329_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_5330_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_re(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ) ).

% suminf_add
tff(fact_5331_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_rj(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ) ).

% suminf_mult2
tff(fact_5332_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_rk(fun(nat,A),fun(A,fun(nat,A)),F),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F)) ) ) ) ).

% suminf_mult
tff(fact_5333_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => ( aa(A,A,minus_minus(A,suminf(A,F)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ) ).

% suminf_diff
tff(fact_5334_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_ra(fun(nat,A),fun(A,fun(nat,A)),F),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),suminf(A,F)),C2) ) ) ) ).

% suminf_divide
tff(fact_5335_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( suminf(A,aTP_Lamp_rf(fun(nat,A),fun(nat,A),F)) = uminus_uminus(A,suminf(A,F)) ) ) ) ).

% suminf_minus
tff(fact_5336_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => summable(B,aa(A,fun(nat,B),F,I2)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_rm(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_rp(fun(A,fun(nat,B)),fun(A,B),F)),I5) ) ) ) ).

% suminf_sum
tff(fact_5337_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_rq(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_rq(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_5338_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N))
           => ( ( suminf(A,F) = zero_zero(A) )
            <=> ! [N6: nat] : aa(nat,A,F,N6) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_5339_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F)) ) ) ) ).

% suminf_nonneg
tff(fact_5340_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F)) ) ) ) ).

% suminf_pos
tff(fact_5341_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A)] : summable(A,aTP_Lamp_rt(fun(nat,A),fun(nat,A),F)) ) ).

% summable_0_powser
tff(fact_5342_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F: fun(nat,A)] : summable(A,aTP_Lamp_ru(fun(nat,A),fun(nat,A),F)) ) ).

% summable_zero_power'
tff(fact_5343_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_rv(fun(nat,A),fun(A,fun(nat,A)),F),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_rw(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ).

% powser_split_head(3)
tff(fact_5344_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_rx(fun(nat,A),fun(A,fun(nat,A)),F),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_ix(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ).

% summable_powser_split_head
tff(fact_5345_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ry(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_ix(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_5346_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),G: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_rz(fun(nat,A),fun(nat,real),F)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_5347_summable__rabs__comparison__test,axiom,
    ! [F: fun(nat,real),G: fun(nat,real)] :
      ( ? [N9: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F,N))),aa(nat,real,G,N)) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_rh(fun(nat,real),fun(nat,real),F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_5348_summable__rabs,axiom,
    ! [F: fun(nat,real)] :
      ( summable(real,aTP_Lamp_rh(fun(nat,real),fun(nat,real),F))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F))),suminf(real,aTP_Lamp_rh(fun(nat,real),fun(nat,real),F))) ) ).

% summable_rabs
tff(fact_5349_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F))
            <=> ? [I3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F,I3)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_5350_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),I: nat] :
          ( summable(A,F)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F,I))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F)) ) ) ) ) ).

% suminf_pos2
tff(fact_5351_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Xa: A] :
          ( summable(A,F)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),N))),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F)),Xa) ) ) ) ).

% suminf_le_const
tff(fact_5352_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_rv(fun(nat,A),fun(A,fun(nat,A)),F),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_rv(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ) ).

% powser_inside
tff(fact_5353_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( suminf(A,aTP_Lamp_rg(fun(nat,A),fun(nat,A),F)) = aa(A,A,minus_minus(A,suminf(A,F)),aa(nat,A,F,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_5354_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Xa: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),N))),Xa)
           => summable(A,F) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_5355_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B4: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),aa(nat,set(nat),set_ord_atMost(nat),N))),B4)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_5356_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),Xa)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_5357_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(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_5358_VEBT__internal_Oreplicatei_Osimps_I1_J,axiom,
    ! [A: $tType,Xa: heap_Time_Heap(A)] : vEBT_VEBT_replicatei(A,zero_zero(nat),Xa) = heap_Time_return(list(A),nil(A)) ).

% VEBT_internal.replicatei.simps(1)
tff(fact_5359_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ri(fun(nat,A),fun(nat,real),F))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F))),suminf(real,aTP_Lamp_ri(fun(nat,A),fun(nat,real),F))) ) ) ).

% summable_norm
tff(fact_5360_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),I5: set(nat)] :
          ( summable(A,F)
         => ( aa(set(nat),$o,finite_finite2(nat),I5)
           => ( ! [N: nat] :
                  ( aa(set(nat),$o,member(nat,N),uminus_uminus(set(nat),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),I5)),suminf(A,F)) ) ) ) ) ).

% sum_le_suminf
tff(fact_5361_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,F)
         => ( suminf(A,F) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),K))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_5362_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,F)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),K)) = aa(A,A,minus_minus(A,suminf(A,F)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_5363_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Nb: nat] :
          ( summable(A,F)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F,M2)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F)) ) ) ) ).

% sum_less_suminf
tff(fact_5364_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N8: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),M5)
                 => ! [N6: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,M5,N6)))),E3) ) ) ) ) ).

% summable_Cauchy
tff(fact_5365_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_rv(fun(nat,A),fun(A,fun(nat,A)),F),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_rv(fun(nat,A),fun(A,fun(nat,A)),F),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_rw(fun(nat,A),fun(A,fun(nat,A)),F),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_5366_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_rv(fun(nat,A),fun(A,fun(nat,A)),F),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_rw(fun(nat,A),fun(A,fun(nat,A)),F),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_rv(fun(nat,A),fun(A,fun(nat,A)),F),Z))),aa(nat,A,F,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_5367_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_5368_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$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_5369_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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_5370_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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_5371_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( summable(A,F)
           => ? [N10: nat] :
              ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),N2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),N2)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_5372_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A),E2: real] :
          ( summable(A,F)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ~ ! [N10: nat] :
                  ~ ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),M3)
                     => ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,M3,N2)))),E2) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_5373_summable__power__series,axiom,
    ! [F: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F,I2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_sa(fun(nat,real),fun(real,fun(nat,real)),F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_5374_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),R0)
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N))),M6)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_sb(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_5375_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_rq(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_rq(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_rs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_5376_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N7: nat,F: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F,N)))) )
           => summable(A,F) ) ) ) ).

% summable_ratio_test
tff(fact_5377_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Nb: nat,I: nat] :
          ( summable(A,F)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,M2)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F,I))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_5378_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,minus_minus(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5379_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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,$o,aa(int,fun(int,$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_5380_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(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,$o,aa(int,fun(int,$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_5381_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),uminus_uminus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Nb)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5382_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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_5383_sum__pos__lt__pair,axiom,
    ! [F: fun(nat,real),K: nat] :
      ( summable(real,F)
     => ( ! [D2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)))),aa(nat,real,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(real,F)) ) ) ).

% sum_pos_lt_pair
tff(fact_5384_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_5385_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_5386_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( tanh(real,ln_ln(real,Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_5387_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2638667681897837118et_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_5388_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_5389_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2638667681897837118et_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_5390_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_5391_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( tanh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_5392_tanh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( tanh(real,Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_5393_tanh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,Xa)),tanh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% tanh_real_le_iff
tff(fact_5394_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),tanh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% tanh_real_pos_iff
tff(fact_5395_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),tanh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_5396_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_5397_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),tanh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_5398_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_se2638667681897837118et_bit(int,Nb,K)),K) ).

% unset_bit_less_eq
tff(fact_5399_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,M,A2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_5400_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_5401_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_5402_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) != zero_zero(A) )
           => ( tan(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),tan(A,Xa))),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),tan(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_5403_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,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),uminus_uminus(code_integer,one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_5404_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
           => ( aa(real,$o,aa(real,fun(real,$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_5405_tan__pi,axiom,
    tan(real,pi) = zero_zero(real) ).

% tan_pi
tff(fact_5406_tan__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( tan(A,zero_zero(A)) = zero_zero(A) ) ) ).

% tan_zero
tff(fact_5407_tan__npi,axiom,
    ! [Nb: nat] : tan(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_5408_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_5409_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_5410_zero__complex_Ocode,axiom,
    zero_zero(complex) = complex2(zero_zero(real),zero_zero(real)) ).

% zero_complex.code
tff(fact_5411_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = uminus_uminus(complex,aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A2 = uminus_uminus(real,aa(num,real,numeral_numeral(real),W)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_5412_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_5413_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_5414_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_5415_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_5416_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_5417_uminus__integer__code_I1_J,axiom,
    uminus_uminus(code_integer,zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_5418_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_5419_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_5420_Complex__sum_H,axiom,
    ! [A: $tType,F: fun(A,real),S2: set(A)] : aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7311177749621191930dd_sum(A,complex),aTP_Lamp_sc(fun(A,real),fun(A,complex),F)),S2) = complex2(aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),F),S2),zero_zero(real)) ).

% Complex_sum'
tff(fact_5421_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = uminus_uminus(complex,one_one(complex)) )
    <=> ( ( A2 = uminus_uminus(real,one_one(real)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_5422_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),tan(real,X)) ) ) ).

% lemma_tan_total
tff(fact_5423_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),tan(real,Xa)) ) ) ).

% tan_gt_zero
tff(fact_5424_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),tan(A,Xa)),tan(A,Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_5425_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),tan(real,Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_5426_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( tan(real,X) = Y ) ) ) ).

% tan_total_pos
tff(fact_5427_tan__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),uminus_uminus(real,pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),tan(real,Xa)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_5428_tan__mono__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tan(real,Xa)),tan(real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_5429_tan__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tan(real,Xa)),tan(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_5430_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),uminus_uminus(code_integer,K),K) ).

% abs_integer_code
tff(fact_5431_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_5432_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_5433_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_5434_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) != zero_zero(A) )
             => ( tan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),tan(A,Xa)),tan(A,Y))),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),tan(A,Xa)),tan(A,Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_5435_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,Xa),Y)) != zero_zero(A) )
             => ( tan(A,aa(A,A,minus_minus(A,Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,tan(A,Xa)),tan(A,Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),tan(A,Xa)),tan(A,Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_5436_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),tan(A,Xa)),tan(A,Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_5437_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,minus_minus(code_integer,zero_zero(code_integer)),L) = uminus_uminus(code_integer,L) ).

% minus_integer_code(2)
tff(fact_5438_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_5439_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : tan(A,Xa) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa))),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))),Xa))),one_one(A))) ) ).

% tan_half
tff(fact_5440_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        uminus_uminus(code_integer,code_integer_of_int(uminus_uminus(int,K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
            $ite(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_5441_sin__paired,axiom,
    ! [Xa: real] : aa(real,$o,sums(real,aTP_Lamp_sd(real,fun(nat,real),Xa)),sin(real,Xa)) ).

% sin_paired
tff(fact_5442_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,log(B2,Xa)) = 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,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),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_5443_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_5444_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_5445_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_qx(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_5446_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A] :
          powr(A,Xa,zero_zero(A)) = $ite(Xa = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_5447_powr__gt__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,Xa,A2))
    <=> ( Xa != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_5448_powr__nonneg__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,A2,Xa)),zero_zero(real))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_5449_powr__eq__one__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( ( powr(real,A2,Xa) = one_one(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_5450_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( powr(real,Xa,one_one(real)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_5451_powr__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,one_one(real)) = Xa ) ) ).

% powr_one
tff(fact_5452_powr__le__cancel__iff,axiom,
    ! [Xa: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Xa,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_5453_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(A2,powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_5454_powr__log__cancel,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( powr(real,A2,log(A2,Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_5455_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xa: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_rt(fun(nat,A),fun(nat,A),A2)),Xa)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xa ) ) ) ).

% powser_sums_zero_iff
tff(fact_5456_powr__numeral,axiom,
    ! [Xa: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(num,real,numeral_numeral(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),Nb)) ) ) ).

% powr_numeral
tff(fact_5457_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aTP_Lamp_rf(fun(nat,A),fun(nat,A),F)),uminus_uminus(A,A2)) ) ) ).

% sums_minus
tff(fact_5458_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_5459_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F: fun(nat,A)] :
          ( ! [N: nat] : aa(nat,A,F,N) = zero_zero(A)
         => aa(A,$o,sums(A,F),zero_zero(A)) ) ) ).

% sums_0
tff(fact_5460_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F: fun(nat,A)] : aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qw(nat,fun(fun(nat,A),fun(nat,A)),I),F)),aa(nat,A,F,I)) ) ).

% sums_single
tff(fact_5461_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% sums_diff
tff(fact_5462_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F: fun(A,fun(nat,B)),Xa: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),F,I2)),aa(A,B,Xa,I2)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_rm(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xa),I5)) ) ) ).

% sums_sum
tff(fact_5463_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_rk(fun(nat,A),fun(A,fun(nat,A)),F),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ).

% sums_mult
tff(fact_5464_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_rj(fun(nat,A),fun(A,fun(nat,A)),F),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ).

% sums_mult2
tff(fact_5465_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ra(fun(nat,A),fun(A,fun(nat,A)),F),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ).

% sums_divide
tff(fact_5466_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),G: fun(nat,A),S2: A,Ta: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N)),aa(nat,A,G,N))
         => ( aa(A,$o,sums(A,F),S2)
           => ( aa(A,$o,sums(A,G),Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S2),Ta) ) ) ) ) ).

% sums_le
tff(fact_5467_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_re(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_5468_powr__mono,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Xa,B2)) ) ) ).

% powr_mono
tff(fact_5469_powr__mono2,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mono2
tff(fact_5470_powr__ge__pzero,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,Xa,Y)) ).

% powr_ge_pzero
tff(fact_5471_powr__less__mono2__neg,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Y,A2)),powr(real,Xa,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_5472_powr__non__neg,axiom,
    ! [A2: real,Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A2,Xa)),zero_zero(real)) ).

% powr_non_neg
tff(fact_5473_zero__integer__def,axiom,
    zero_zero(code_integer) = code_integer_of_int(zero_zero(int)) ).

% zero_integer_def
tff(fact_5474_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_se(A,fun(fun(nat,A),fun(nat,A)),C2),F)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D))
          <=> aa(A,$o,sums(A,F),D) ) ) ) ).

% sums_mult_iff
tff(fact_5475_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sf(A,fun(fun(nat,A),fun(nat,A)),C2),F)),aa(A,A,aa(A,fun(A,A),times_times(A),D),C2))
          <=> aa(A,$o,sums(A,F),D) ) ) ) ).

% sums_mult2_iff
tff(fact_5476_powr__mono2_H,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A2)),powr(real,Xa,A2)) ) ) ) ).

% powr_mono2'
tff(fact_5477_powr__less__mono2,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_5478_powr__inj,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,Xa) = powr(real,A2,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% powr_inj
tff(fact_5479_gr__one__powr,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),powr(real,Xa,Y)) ) ) ).

% gr_one_powr
tff(fact_5480_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,Xa,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_5481_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_5482_powr__le1,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_5483_powr__divide,axiom,
    ! [Xa: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_5484_powr__mult,axiom,
    ! [Xa: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_5485_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qz(A,fun(fun(nat,A),fun(nat,A)),C2),F)),A2)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% sums_mult_D
tff(fact_5486_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),S2: A] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_rg(fun(nat,A),fun(nat,A),F)),S2)
           => aa(A,$o,sums(A,F),S2) ) ) ) ).

% sums_Suc_imp
tff(fact_5487_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),L: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_sg(fun(nat,A),fun(nat,A),F)),L)
         => aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_5488_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),S2: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_rg(fun(nat,A),fun(nat,A),F)),S2)
        <=> aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(nat,A,F,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_5489_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F: fun(nat,A),S2: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
             => ( aa(nat,A,F,I2) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sh(nat,fun(fun(nat,A),fun(nat,A)),Nb),F)),S2)
          <=> aa(A,$o,sums(A,F),S2) ) ) ) ).

% sums_zero_iff_shift
tff(fact_5490_log__base__powr,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( ( A2 != zero_zero(real) )
     => ( log(powr(real,A2,B2),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),log(A2,Xa)),B2) ) ) ).

% log_base_powr
tff(fact_5491_log__powr,axiom,
    ! [Xa: real,B2: real,Y: real] :
      ( ( Xa != zero_zero(real) )
     => ( log(B2,powr(real,Xa,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),log(B2,Xa)) ) ) ).

% log_powr
tff(fact_5492_ln__powr,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != zero_zero(real) )
     => ( ln_ln(real,powr(real,Xa,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),ln_ln(real,Xa)) ) ) ).

% ln_powr
tff(fact_5493_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xa: A,A2: A,B2: A] : powr(A,Xa,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,Xa,A2)),powr(A,Xa,B2)) ) ).

% powr_add
tff(fact_5494_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N7: set(nat),F: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N7)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),N7)
               => ( aa(nat,A,F,N) = zero_zero(A) ) )
           => aa(A,$o,sums(A,F),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),N7)) ) ) ) ).

% sums_finite
tff(fact_5495_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rb(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(fun(nat,$o),set(nat),collect(nat),P))) ) ) ).

% sums_If_finite
tff(fact_5496_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A3)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rc(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),A3)) ) ) ).

% sums_If_finite_set
tff(fact_5497_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_si(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_5498_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_rt(fun(nat,A),fun(nat,A),A2)),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_5499_powr__realpow,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(nat,real,semiring_1_of_nat(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Nb) ) ) ).

% powr_realpow
tff(fact_5500_less__log__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),log(B2,Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xa) ) ) ) ).

% less_log_iff
tff(fact_5501_log__less__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),log(B2,Xa)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_5502_less__powr__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),log(B2,Xa)),Y) ) ) ) ).

% less_powr_iff
tff(fact_5503_powr__less__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),log(B2,Xa)) ) ) ) ).

% powr_less_iff
tff(fact_5504_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),Nb: nat,S2: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),Nb)),S2)
        <=> aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_iff_shift
tff(fact_5505_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),Nb: nat,S2: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),Nb)),aa(A,A,minus_minus(A,S2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),Nb))))
        <=> aa(A,$o,sums(A,F),S2) ) ) ).

% sums_iff_shift'
tff(fact_5506_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),S2: A,Nb: nat] :
          ( aa(A,$o,sums(A,F),S2)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_qy(fun(nat,A),fun(nat,fun(nat,A)),F),Nb)),aa(A,A,minus_minus(A,S2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_5507_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S3: A,A3: set(nat),S6: A,F: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),S3)
         => ( aa(set(nat),$o,finite_finite2(nat),A3)
           => ( ( S6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_sj(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F)),A3)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_sk(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F)),S6) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_5508_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,uminus_uminus(real,one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xa) ) ) ).

% powr_neg_one
tff(fact_5509_powr__mult__base,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,Xa,Y)) = powr(real,Xa,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_5510_powr__le__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),log(B2,Xa)) ) ) ) ).

% powr_le_iff
tff(fact_5511_le__powr__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(B2,Xa)),Y) ) ) ) ).

% le_powr_iff
tff(fact_5512_log__le__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(B2,Xa)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_5513_le__log__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),log(B2,Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xa) ) ) ) ).

% le_log_iff
tff(fact_5514_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F: fun(nat,A),S2: A,K: nat] :
          ( aa(A,$o,sums(A,F),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_sl(fun(nat,A),fun(nat,fun(nat,A)),F),K)),S2) ) ) ) ).

% sums_group
tff(fact_5515_ln__powr__bound,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ln_ln(real,Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xa,A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_5516_ln__powr__bound2,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,ln_ln(real,Xa),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_5517_add__log__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),log(B2,Xa)) = log(B2,aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_5518_log__add__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),log(B2,Xa)),Y) = log(B2,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_5519_minus__log__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,minus_minus(real,Y),log(B2,Xa)) = log(B2,aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_5520_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A,A2: A] :
          powr(A,Xa,A2) = $ite(Xa = zero_zero(A),zero_zero(A),exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),ln_ln(A,Xa)))) ) ).

% powr_def
tff(fact_5521_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),power_power(A),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_5522_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_sm(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_5523_log__minus__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,minus_minus(real,log(B2,Xa)),Y) = log(B2,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B2,uminus_uminus(real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_5524_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_rq(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_rq(fun(nat,A),fun(nat,real),B2))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rs(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_5525_sums__if_H,axiom,
    ! [G: fun(nat,real),Xa: real] :
      ( aa(real,$o,sums(real,G),Xa)
     => aa(real,$o,sums(real,aTP_Lamp_sn(fun(nat,real),fun(nat,real),G)),Xa) ) ).

% sums_if'
tff(fact_5526_sums__if,axiom,
    ! [G: fun(nat,real),Xa: real,F: fun(nat,real),Y: real] :
      ( aa(real,$o,sums(real,G),Xa)
     => ( aa(real,$o,sums(real,F),Y)
       => aa(real,$o,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_so(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)) ) ) ).

% sums_if
tff(fact_5527_powr__neg__numeral,axiom,
    ! [Xa: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,uminus_uminus(real,aa(num,real,numeral_numeral(real),Nb))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),Nb))) ) ) ).

% powr_neg_numeral
tff(fact_5528_powr__int,axiom,
    ! [Xa: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),nat2(I)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),nat2(uminus_uminus(int,I))))) ) ) ).

% powr_int
tff(fact_5529_cos__paired,axiom,
    ! [Xa: real] : aa(real,$o,sums(real,aTP_Lamp_sp(real,fun(nat,real),Xa)),cos(real,Xa)) ).

% cos_paired
tff(fact_5530_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => aa(A,$o,sums(A,aTP_Lamp_sq(A,fun(nat,A),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_5531_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ss(A,fun(A,fun(nat,A)),Xa),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xa)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_5532_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_su(A,fun(A,fun(nat,A)),Xa),Y)),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_5533_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_sw(A,fun(A,fun(nat,A)),Xa),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_5534_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_5535_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xa: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa) )
        <=> ( ( A2 = B2 )
            | ( Xa = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_5536_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A2: real,Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)) ) ).

% mult_scaleR_left
tff(fact_5537_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [Xa: A,A2: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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),Xa),Y)) ) ).

% mult_scaleR_right
tff(fact_5538_scaleR__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xa: A,Y: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
        <=> ( ( Xa = Y )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_cancel_left
tff(fact_5539_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),Xa) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_5540_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xa: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( Xa = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_5541_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_5542_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Va2: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),Va2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),W)),aa(num,real,numeral_numeral(real),Va2))),A2) ) ).

% inverse_scaleR_times
tff(fact_5543_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,Va2: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),Va2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),aa(num,real,numeral_numeral(real),Va2))),A2) ) ).

% fraction_scaleR_times
tff(fact_5544_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),A2: A,R2: real] :
          ( aa(A,$o,sums(A,X5),A2)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_sx(fun(nat,A),fun(real,fun(nat,A)),X5),R2)),aa(A,A,real_V8093663219630862766scaleR(A,R2),A2)) ) ) ).

% sums_scaleR_right
tff(fact_5545_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,F: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_sy(real,fun(fun(B,A),fun(B,A)),A2),F)),A3) ) ).

% scaleR_sum_right
tff(fact_5546_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,G: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_sy(real,fun(fun(B,A),fun(B,A)),A2),G)),A3) ) ).

% scaleR_right.sum
tff(fact_5547_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R2: real] :
          ( summable(A,X5)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_sx(fun(nat,A),fun(real,fun(nat,A)),X5),R2)) ) ) ).

% summable_scaleR_right
tff(fact_5548_scaleR__left__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xa: A,Y: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
           => ( Xa = Y ) ) ) ) ).

% scaleR_left_imp_eq
tff(fact_5549_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,A2: real,B2: real] :
          ( ( Xa != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_5550_scaleR__left_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(B,real),A3: set(B),Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),G),A3)),Xa) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_sz(fun(B,real),fun(A,fun(B,A)),G),Xa)),A3) ) ).

% scaleR_left.sum
tff(fact_5551_scaleR__sum__left,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [F: fun(B,real),A3: set(B),Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),F),A3)),Xa) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_sz(fun(B,real),fun(A,fun(B,A)),F),Xa)),A3) ) ).

% scaleR_sum_left
tff(fact_5552_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R2: real] :
          ( summable(A,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R2),suminf(A,X5)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_sx(fun(nat,A),fun(real,fun(nat,A)),X5),R2)) ) ) ) ).

% suminf_scaleR_right
tff(fact_5553_summable__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),Xa: A] :
          ( summable(real,X5)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ta(fun(nat,real),fun(A,fun(nat,A)),X5),Xa)) ) ) ).

% summable_scaleR_left
tff(fact_5554_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),A2: real,Xa: A] :
          ( aa(real,$o,sums(real,X5),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ta(fun(nat,real),fun(A,fun(nat,A)),X5),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)) ) ) ).

% sums_scaleR_left
tff(fact_5555_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_5556_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa)) ) ) ) ).

% scaleR_right_mono
tff(fact_5557_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_5558_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_5559_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_5560_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xa: A,Y: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_5561_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_5562_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,Va2: real,A2: A,Xa: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),Va2)),A2) = Xa )
        <=> $ite(Va2 = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,Va2),Xa)) ) ) ).

% vector_fraction_eq_iff
tff(fact_5563_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,U: real,Va2: real,A2: A] :
          ( ( Xa = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),Va2)),A2) )
        <=> $ite(Va2 = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,Va2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_5564_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,B2),A2)),E2)),D)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_5565_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),E2)),C2)),D) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_5566_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_5567_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_5568_suminf__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),Xa: A] :
          ( summable(real,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,suminf(real,X5)),Xa) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ta(fun(nat,real),fun(A,fun(nat,A)),X5),Xa)) ) ) ) ).

% suminf_scaleR_left
tff(fact_5569_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_5570_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_5571_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_5572_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_5573_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_5574_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_5575_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_5576_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_5577_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xa: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),Xa) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_5578_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,Xa,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),Xa),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_5579_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_tb(A,fun(nat,A),Xa)),sin(A,Xa)) ) ).

% sin_converges
tff(fact_5580_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : sin(A,X4) = suminf(A,aTP_Lamp_tb(A,fun(nat,A),X4)) ) ).

% sin_def
tff(fact_5581_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_tc(A,fun(nat,A),Xa)),cos(A,Xa)) ) ).

% cos_converges
tff(fact_5582_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : cos(A,X4) = suminf(A,aTP_Lamp_tc(A,fun(nat,A),X4)) ) ).

% cos_def
tff(fact_5583_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(real,aTP_Lamp_td(A,fun(nat,real),Xa)) ) ).

% summable_norm_sin
tff(fact_5584_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(real,aTP_Lamp_te(A,fun(nat,real),Xa)) ) ).

% summable_norm_cos
tff(fact_5585_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_tf(A,fun(nat,A),Xa)),sin(A,Xa)) ) ).

% sin_minus_converges
tff(fact_5586_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_tg(A,fun(nat,A),Xa)),cos(A,Xa)) ) ).

% cos_minus_converges
tff(fact_5587_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_th(fun(nat,A),fun(A,fun(nat,A)),C2),Xa))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ti(fun(nat,A),fun(A,fun(nat,A)),C2),Xa)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_th(fun(nat,A),fun(A,fun(nat,A)),C2),Xa))) ) ) ).

% diffs_equiv
tff(fact_5588_time__array__map__entry,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,F: fun(A,A),P3: array(A),H: heap_ext(product_unit)] :
          time_time(array(A),array_map_entry(A,I,F,P3),H) = $ite(time_fails(array(A),array_map_entry(A,I,F,P3),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_map_entry
tff(fact_5589_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M2)),aa(nat,A,X5,N)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_5590_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : diffs(A,aTP_Lamp_tj(fun(nat,A),fun(nat,A),C2),X4) = uminus_uminus(A,diffs(A,C2,X4)) ) ).

% diffs_minus
tff(fact_5591_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : 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_5592_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xa: A] :
          ( ! [X: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_tk(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_tl(fun(nat,A),fun(A,fun(nat,A)),C2),Xa)) ) ) ).

% termdiff_converges_all
tff(fact_5593_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,K4: real,C2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),K4)
         => ( ! [X: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),K4)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_tk(fun(nat,A),fun(A,fun(nat,A)),C2),X)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tm(A,fun(fun(nat,A),fun(nat,A)),Xa),C2)) ) ) ) ).

% termdiff_converges
tff(fact_5594_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: fun(nat,A)] :
          ( topological_monoseq(A,A2)
         => topological_monoseq(A,aTP_Lamp_tn(fun(nat,A),fun(nat,A),A2)) ) ) ).

% monoseq_minus
tff(fact_5595_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [N6: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N6)),aa(nat,A,X5,aa(nat,nat,suc,N6)))
            | ! [N6: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N6))),aa(nat,A,X5,N6)) ) ) ) ).

% monoseq_Suc
tff(fact_5596_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI2
tff(fact_5597_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI1
tff(fact_5598_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [M5: nat,N6: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M5)),aa(nat,A,X5,N6)) )
            | ! [M5: nat,N6: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N6)),aa(nat,A,X5,M5)) ) ) ) ) ).

% monoseq_def
tff(fact_5599_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,M2)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_5600_time__array__swap,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xa: A,P3: array(A),H: heap_ext(product_unit),F: fun(A,A)] :
          time_time(A,array_swap(A,I,Xa,P3),H) = $ite(time_fails(array(A),array_map_entry(A,I,F,P3),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_swap
tff(fact_5601_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : exp(A,Xa) = 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)),Xa)),suminf(A,aTP_Lamp_to(A,fun(nat,A),Xa))) ) ).

% exp_first_two_terms
tff(fact_5602_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,log(aa(nat,real,semiring_1_of_nat(real),B2),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$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_5603_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_5604_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_5605_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_5606_floor__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archim6421214686448440834_floor(A,aa(int,A,ring_1_of_int(A),Z)) = Z ) ).

% floor_of_int
tff(fact_5607_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_5608_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_5609_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% inverse_positive_iff_positive
tff(fact_5610_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_5611_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_5612_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_5613_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_5614_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),Va2)) = aa(num,int,numeral_numeral(int),Va2) ) ).

% floor_numeral
tff(fact_5615_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_5616_floor__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: nat] : archim6421214686448440834_floor(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ).

% floor_of_nat
tff(fact_5617_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_5618_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_5619_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_5620_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_5621_floor__uminus__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archim6421214686448440834_floor(A,uminus_uminus(A,aa(int,A,ring_1_of_int(A),Z))) = uminus_uminus(int,Z) ) ).

% floor_uminus_of_int
tff(fact_5622_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xa),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xa)),Z) ) ).

% floor_diff_of_int
tff(fact_5623_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ) ).

% zero_le_floor
tff(fact_5624_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Va2)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Va2)),Xa) ) ) ).

% numeral_le_floor
tff(fact_5625_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_5626_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),Va2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),Va2)) ) ) ).

% floor_less_numeral
tff(fact_5627_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% zero_less_floor
tff(fact_5628_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_5629_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% one_le_floor
tff(fact_5630_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_less_one
tff(fact_5631_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num] : archim6421214686448440834_floor(A,uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) = uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2)) ) ).

% floor_neg_numeral
tff(fact_5632_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xa),aa(num,A,numeral_numeral(A),Va2))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),Va2)) ) ).

% floor_diff_numeral
tff(fact_5633_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xa),one_one(A))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xa)),one_one(int)) ) ).

% floor_diff_one
tff(fact_5634_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Va2)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Va2)),one_one(A))),Xa) ) ) ).

% numeral_less_floor
tff(fact_5635_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),Va2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Va2)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_5636_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa) ) ) ).

% one_less_floor
tff(fact_5637_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_5638_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),Xa) ) ) ).

% neg_numeral_le_floor
tff(fact_5639_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) ) ) ).

% floor_less_neg_numeral
tff(fact_5640_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Va2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),one_one(A))),Xa) ) ) ).

% neg_numeral_less_floor
tff(fact_5641_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Va2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Va2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_5642_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),uminus_uminus(A,A2)) = uminus_uminus(A,aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_5643_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: int,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ).

% mult_inverse_of_int_commute
tff(fact_5644_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_5645_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),real_V7770717601297561774m_norm(A,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xa))),aa(real,real,inverse_inverse(real),R2)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_5646_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa) ) ).

% of_int_floor_le
tff(fact_5647_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_5648_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xa: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_5649_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_5650_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_5651_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: nat,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_5652_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: 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),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),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),Xa)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_5653_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: 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),Xa),M)),aa(A,A,inverse_inverse(A),Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)) ) ).

% power_mult_inverse_distrib
tff(fact_5654_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A2) ) ).

% divide_inverse_commute
tff(fact_5655_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_5656_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_5657_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% floor_le_ceiling
tff(fact_5658_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_5659_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_5660_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_5661_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_5662_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_5663_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_5664_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less
tff(fact_5665_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(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_5666_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_5667_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_5668_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_5669_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_5670_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_5671_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ).

% positive_imp_inverse_positive
tff(fact_5672_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% floor_less_cancel
tff(fact_5673_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,Xa: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( Xa != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),Xa)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_5674_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_round(A,Xa)) ) ).

% floor_le_round
tff(fact_5675_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: nat] : diffs(A,aTP_Lamp_tp(nat,A),X4) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X4)) ) ).

% exp_fdiffs
tff(fact_5676_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_5677_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_5678_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_5679_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_5680_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% inverse_le_1_iff
tff(fact_5681_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_5682_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_less_inverse
tff(fact_5683_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% le_floor_iff
tff(fact_5684_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_5685_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_5686_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_5687_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_5688_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_5689_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_5690_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_5691_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,Xa)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),Xa)) ) ).

% int_add_floor
tff(fact_5692_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))) ) ).

% le_floor_add
tff(fact_5693_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) ) ).

% floor_divide_of_int_eq
tff(fact_5694_ceiling__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,uminus_uminus(A,Xa)) = uminus_uminus(int,archim6421214686448440834_floor(A,Xa)) ) ).

% ceiling_minus
tff(fact_5695_floor__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archim6421214686448440834_floor(A,uminus_uminus(A,Xa)) = uminus_uminus(int,archimedean_ceiling(A,Xa)) ) ).

% floor_minus
tff(fact_5696_ceiling__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,Xa) = uminus_uminus(int,archim6421214686448440834_floor(A,uminus_uminus(A,Xa))) ) ).

% ceiling_def
tff(fact_5697_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A2) = aa(real,real,inverse_inverse(real),powr(real,Y,A2)) ) ) ).

% inverse_powr
tff(fact_5698_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
         => aa(A,$o,aa(A,fun(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_5699_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_5700_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% inverse_less_1_iff
tff(fact_5701_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_le_inverse
tff(fact_5702_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_5703_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_5704_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),one_one(A))) ) ).

% one_add_floor
tff(fact_5705_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)) = 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_5706_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,Nb: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)) ) ).

% floor_divide_of_nat_eq
tff(fact_5707_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),Xa) ) ) ).

% reals_Archimedean
tff(fact_5708_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(archim6421214686448440834_floor(A,A2))),nat2(archim6421214686448440834_floor(A,B2)))),nat2(archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ).

% le_mult_nat_floor
tff(fact_5709_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( nat2(archim6421214686448440834_floor(real,Xa)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_5710_floor__eq3,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( nat2(archim6421214686448440834_floor(real,Xa)) = Nb ) ) ) ).

% floor_eq3
tff(fact_5711_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_ceiling(A,Xa) = $ite(Xa = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Xa),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_5712_le__nat__floor,axiom,
    ! [Xa: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xa)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),nat2(archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_5713_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,archimedean_ceiling(A,Xa)),archim6421214686448440834_floor(A,Xa))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_5714_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Xa: 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),Xa)),C2) = Y )
          <=> ( Xa = 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_5715_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,Xa: 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),Xa)),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)) = Xa ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_5716_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_5717_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_5718_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_5719_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_5720_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_5721_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_5722_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_5723_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_5724_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$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_5725_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D2: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D2),E)
         => ( aa(real,$o,P,D2)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_5726_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
    <=> ? [N6: nat] :
          ( ( N6 != zero_zero(nat) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N6)))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N6))),E2) ) ) ).

% real_arch_inverse
tff(fact_5727_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D2: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D2),E)
         => ( aa(real,$o,P,D2)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_5728_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$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_5729_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ln_ln(real,aa(real,real,inverse_inverse(real),Xa)) = uminus_uminus(real,ln_ln(real,Xa)) ) ) ).

% ln_inverse
tff(fact_5730_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : summable(A,aTP_Lamp_tq(A,fun(nat,A),Xa)) ) ).

% summable_exp
tff(fact_5731_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I3)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))) )
             => aa(int,$o,P,I3) ) ) ) ).

% floor_split
tff(fact_5732_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xa) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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_5733_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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,Xa) = Z ) ) ) ) ).

% floor_unique
tff(fact_5734_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% less_floor_iff
tff(fact_5735_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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_5736_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% le_mult_floor
tff(fact_5737_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(A,aTP_Lamp_tr(A,fun(nat,A),Xa)) ) ).

% summable_exp_generic
tff(fact_5738_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_5739_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),Xa) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_5740_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: nat,Nb: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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),Xa),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_5741_floor__eq4,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( nat2(archim6421214686448440834_floor(real,Xa)) = Nb ) ) ) ).

% floor_eq4
tff(fact_5742_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),uminus_uminus(A,B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_5743_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_5744_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_5745_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),uminus_uminus(A,B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_5746_floor__eq2,axiom,
    ! [Nb: int,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Nb)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),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,Xa) = Nb ) ) ) ).

% floor_eq2
tff(fact_5747_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),uminus_uminus(A,B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_5748_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_5749_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),uminus_uminus(A,B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_5750_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),uminus_uminus(A,B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_5751_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_5752_log__inverse,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( log(A2,aa(real,real,inverse_inverse(real),Xa)) = uminus_uminus(real,log(A2,Xa)) ) ) ) ) ).

% log_inverse
tff(fact_5753_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q4)))),Q4)),P3) ) ) ).

% floor_divide_lower
tff(fact_5754_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_tr(A,fun(nat,A),Xa)),exp(A,Xa)) ) ).

% exp_converges
tff(fact_5755_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : exp(A,X4) = suminf(A,aTP_Lamp_tr(A,fun(nat,A),X4)) ) ).

% exp_def
tff(fact_5756_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(real,aTP_Lamp_ts(A,fun(nat,real),Xa)) ) ).

% summable_norm_exp
tff(fact_5757_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa)))) ).

% exp_plus_inverse_exp
tff(fact_5758_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$o,aa(A,fun(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,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q4)))),one_one(A))),Q4)) ) ) ).

% floor_divide_upper
tff(fact_5759_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_round(A,Xa) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_5760_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa))) ) ).

% plus_inverse_ge_2
tff(fact_5761_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_5762_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_5763_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) )
         => ( 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),Xa),Y)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_tt(A,fun(A,fun(nat,fun(nat,A))),Xa),Y),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_5764_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : exp(A,Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_tu(A,fun(nat,A),Xa))) ) ).

% exp_first_term
tff(fact_5765_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),tan(A,Xa)),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,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_5766_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,log(B2,Xa)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),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_5767_powr__real__of__int,axiom,
    ! [Xa: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),nat2(Nb)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),nat2(uminus_uminus(int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_5768_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archim6421214686448440834_floor(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_5769_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A,K: nat] : exp(A,Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_tr(A,fun(nat,A),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_tv(A,fun(nat,fun(nat,A)),Xa),K))) ) ).

% exp_first_terms
tff(fact_5770_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archim6421214686448440834_floor(real,log(aa(nat,real,semiring_1_of_nat(real),B2),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_5771_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,sin(real,Xa)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qu(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Xa)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_5772_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_tw(A,fun(nat,A),Xa)),sinh(A,Xa)) ) ).

% sinh_converges
tff(fact_5773_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_round(A,Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xa)),archimedean_ceiling(A,Xa),archim6421214686448440834_floor(A,Xa)) ) ).

% round_altdef
tff(fact_5774_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_tx(A,fun(nat,A),Xa)),cosh(A,Xa)) ) ).

% cosh_converges
tff(fact_5775_sinh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( sinh(real,Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_5776_sinh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% sinh_real_le_iff
tff(fact_5777_frac__frac,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_frac(A,archimedean_frac(A,Xa)) = archimedean_frac(A,Xa) ) ).

% frac_frac
tff(fact_5778_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_5779_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sinh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% sinh_real_pos_iff
tff(fact_5780_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_5781_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sinh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_5782_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_5783_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_5784_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_5785_cosh__real__ge__1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,Xa)) ).

% cosh_real_ge_1
tff(fact_5786_cosh__real__nonzero,axiom,
    ! [Xa: real] : cosh(real,Xa) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_5787_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : cosh(A,aa(A,A,minus_minus(A,Xa),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xa)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xa)),sinh(A,Y))) ) ).

% cosh_diff
tff(fact_5788_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : sinh(A,aa(A,A,minus_minus(A,Xa),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xa)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xa)),sinh(A,Y))) ) ).

% sinh_diff
tff(fact_5789_sinh__le__cosh__real,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),cosh(real,Xa)) ).

% sinh_le_cosh_real
tff(fact_5790_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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,Xa)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xa)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_5791_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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,Xa)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xa)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_5792_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = 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,Xa))),cosh(A,Xa)) ) ).

% sinh_double
tff(fact_5793_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cosh(real,Xa)) ).

% cosh_real_nonneg
tff(fact_5794_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_5795_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_5796_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( arcosh(real,cosh(real,Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_5797_cosh__real__pos,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cosh(real,Xa)) ).

% cosh_real_pos
tff(fact_5798_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xa)) ) ).

% frac_ge_0
tff(fact_5799_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xa)),one_one(A)) ) ).

% frac_lt_1
tff(fact_5800_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),one_one(A))) = archimedean_frac(A,Xa) ) ).

% frac_1_eq
tff(fact_5801_cosh__real__strict__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_5802_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_5803_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_5804_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_5805_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_frac(A,Xa) = aa(A,A,minus_minus(A,Xa),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))) ) ).

% frac_def
tff(fact_5806_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = Xa )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_5807_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_5808_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cosh(A,Xa) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( tanh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),tanh(A,Xa)),tanh(A,Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),tanh(A,Xa)),tanh(A,Y)))) ) ) ) ) ).

% tanh_add
tff(fact_5809_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sinh(A,Xa) = zero_zero(A) )
        <=> aa(set(A),$o,member(A,exp(A,Xa)),aa(set(A),set(A),insert(A,one_one(A)),aa(set(A),set(A),insert(A,uminus_uminus(A,one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_5810_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),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,Xa)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_5811_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cosh(A,Xa) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = uminus_uminus(A,one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_5812_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( cosh(real,ln_ln(real,Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_5813_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( sinh(real,ln_ln(real,Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,Xa),aa(real,real,inverse_inverse(real),Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_5814_cot__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),uminus_uminus(real,pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cot(real,Xa)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_5815_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W))
          & aa(int,$o,aa(int,fun(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_5816_le__arcsin__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),uminus_uminus(real,pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),arcsin(Xa))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_5817_arcsin__0,axiom,
    arcsin(zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_5818_cot__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( cot(A,zero_zero(A)) = zero_zero(A) ) ) ).

% cot_zero
tff(fact_5819_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_5820_More__Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( ring_1_signed(A,int,Xa) = zero_zero(int) )
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% More_Word.sint_0
tff(fact_5821_Word_Oof__int__sint,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [A2: word(B)] : aa(int,A,ring_1_of_int(A),ring_1_signed(B,int,A2)) = ring_1_signed(B,A,A2) ) ).

% Word.of_int_sint
tff(fact_5822_cot__pi,axiom,
    cot(real,pi) = zero_zero(real) ).

% cot_pi
tff(fact_5823_signed__minus__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( ring_1_signed(B,A,uminus_uminus(word(B),one_one(word(B)))) = uminus_uminus(A,one_one(A)) ) ) ).

% signed_minus_1
tff(fact_5824_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( sin(real,arcsin(Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_5825_cot__npi,axiom,
    ! [Nb: nat] : cot(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_5826_word__eq__iff__signed,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & ring_char_0(B) )
     => ! [Va2: word(A),W: word(A)] :
          ( ( Va2 = W )
        <=> ( ring_1_signed(A,B,Va2) = ring_1_signed(A,B,W) ) ) ) ).

% word_eq_iff_signed
tff(fact_5827_signed__word__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [Va2: word(B),W: word(B)] :
          ( ( ring_1_signed(B,A,Va2) = ring_1_signed(B,A,W) )
         => ( Va2 = W ) ) ) ).

% signed_word_eqI
tff(fact_5828_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_5829_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_5830_arcsin__le__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Xa)),arcsin(Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_5831_arcsin__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( arcsin(uminus_uminus(real,Xa)) = uminus_uminus(real,arcsin(Xa)) ) ) ) ).

% arcsin_minus
tff(fact_5832_arcsin__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Xa)),arcsin(Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_5833_arcsin__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( ( arcsin(Xa) = arcsin(Y) )
        <=> ( Xa = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_5834_sint__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,uminus_uminus(word(A),one_one(word(A)))) = uminus_uminus(int,one_one(int)) ) ) ).

% sint_n1
tff(fact_5835_arcsin__less__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),arcsin(Xa)),arcsin(Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_5836_arcsin__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arcsin(Xa)),arcsin(Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_5837_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( cos(real,arcsin(Xa)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_5838_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))),Xa)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,W)),Xa) ) ) ).

% sint_above_size
tff(fact_5839_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),uminus_uminus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),ring_1_signed(A,int,W)) ) ) ).

% sint_below_size
tff(fact_5840_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arcsin(Y)) ) ) ).

% arcsin_lbound
tff(fact_5841_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_5842_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arcsin(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_5843_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cot(real,Xa)) ) ) ).

% cot_gt_zero
tff(fact_5844_arcsin__sin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( arcsin(sin(real,Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_5845_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arcsin(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,arcsin(Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_5846_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arcsin(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Y)),pi)
          & ( sin(real,arcsin(Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_5847_arcsin__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),uminus_uminus(real,pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Xa)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_5848_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),uminus_uminus(complex,one_one(complex))),Nb) ).

% i_even_power
tff(fact_5849_set__decode__0,axiom,
    ! [Xa: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),nat_set_decode(Xa))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa) ) ).

% set_decode_0
tff(fact_5850_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_5851_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_5852_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_5853_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_5854_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(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_5855_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_5856_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_5857_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_5858_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : aa(A,A,abs_abs(A),aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% abs_bool_eq
tff(fact_5859_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,aa(A,fun(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_5860_div__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),one_one(word(A))),W) = aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))) ) ).

% div_word_one
tff(fact_5861_set__decode__inverse,axiom,
    ! [Nb: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(Nb)) = Nb ).

% set_decode_inverse
tff(fact_5862_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(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_5863_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(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_5864_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_5865_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_5866_div__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),uminus_uminus(word(A),one_one(word(A)))) = aa($o,word(A),zero_neq_one_of_bool(word(A)),W = uminus_uminus(word(A),one_one(word(A)))) ) ).

% div_word_by_minus_1_eq
tff(fact_5867_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_5868_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_5869_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_5870_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,abs_abs(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5871_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_5872_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_5873_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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% sgn_of_nat
tff(fact_5874_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),F: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_ty(fun(A,B),fun(fun(A,$o),fun(A,B)),F),P)),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_5875_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_tz(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F)),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_5876_mod__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(word(A),W,uminus_uminus(word(A),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($o,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),uminus_uminus(word(A),one_one(word(A)))))) ) ).

% mod_word_by_minus_1_eq
tff(fact_5877_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_5878_set__decode__Suc,axiom,
    ! [Nb: nat,Xa: nat] :
      ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Nb)),nat_set_decode(Xa))
    <=> aa(set(nat),$o,member(nat,Nb),nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_5879_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_5880_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_5881_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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% one_mod_2_pow_eq
tff(fact_5882_scast__n1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(B,word(A),uminus_uminus(word(B),one_one(word(B)))) = uminus_uminus(word(A),one_one(word(A))) ) ) ).

% scast_n1
tff(fact_5883_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_5884_scast__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : ring_1_signed(B,word(A),W) = aa(int,word(A),ring_1_of_int(word(A)),ring_1_signed(B,int,W)) ) ).

% scast_eq
tff(fact_5885_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o,Q4: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P3)) = aa($o,A,zero_neq_one_of_bool(A),(Q4)) )
        <=> ( (P3)
          <=> (Q4) ) ) ) ).

% of_bool_eq_iff
tff(fact_5886_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero(complex) ).

% complex_i_not_zero
tff(fact_5887_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_5888_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_5889_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_5890_finite__set__decode,axiom,
    ! [Nb: nat] : aa(set(nat),$o,finite_finite2(nat),nat_set_decode(Nb)) ).

% finite_set_decode
tff(fact_5891_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_5892_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_5893_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_5894_Complex__eq__i,axiom,
    ! [Xa: real,Y: real] :
      ( ( complex2(Xa,Y) = imaginary_unit )
    <=> ( ( Xa = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_5895_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_5896_subset__decode__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% subset_decode_imp_le
tff(fact_5897_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(list($o),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_5898_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A4),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => aa(A,$o,P,A4) )
         => ( ! [A4: A,B3: $o] :
                ( aa(A,$o,P,A4)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))) ) )
           => aa(A,$o,P,A2) ) ) ) ).

% bits_induct
tff(fact_5899_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,$o,aa(nat,fun(nat,$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_5900_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,minus_minus(int,uminus_uminus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_5901_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Nb))) ) ).

% exp_div_exp_eq
tff(fact_5902_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa(set(nat),$o,member(nat,Nb),nat_set_decode(Z))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Z)) = aa(set(nat),set(nat),insert(nat,Nb),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5903_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),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),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Nb),M)))))) ) ).

% divide_int_unfold
tff(fact_5904_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_5905_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))),uminus_uminus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,dvd_dvd(int,L),K)))))) ) ).

% divide_int_def
tff(fact_5906_modulo__int__def,axiom,
    ! [K: int,L: int] :
      modulo_modulo(int,K,L) = $ite(
        L = zero_zero(int),
        K,
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K)))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_5907_set__decode__def,axiom,
    ! [Xa: nat] : nat_set_decode(Xa) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ua(nat,fun(nat,$o),Xa)) ).

% set_decode_def
tff(fact_5908_and__int_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Xaa) = Y )
     => ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa(set(int),$o,member(int,Xa),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int)))))
                  & aa(set(int),$o,member(int,Xaa),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int))))) ),
                  uminus_uminus(int,
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa)
                      & ~ aa(int,$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))),Xa)
                        & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_5909_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( aa(set(int),$o,member(int,K),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int))))) ),
            uminus_uminus(int,
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
                & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
                  & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_5910_and__int_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Xaa) = Y )
     => ( Y = $ite(
            ( aa(set(int),$o,member(int,Xa),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,Xaa),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int))))) ),
            uminus_uminus(int,
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa)
                & ~ aa(int,$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))),Xa)
                  & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_5911_scast__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : ring_1_signed(A,word(A),W) = W ) ).

% scast_id
tff(fact_5912_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_5913_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xa) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_5914_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_5915_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_5916_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_5917_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_5918_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_5919_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_5920_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_5921_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_5922_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: 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,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_5923_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),uminus_uminus(int,aa(num,int,numeral_numeral(int),bit0(Nb)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_5924_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),uminus_uminus(int,aa(num,int,numeral_numeral(int),bit0(Nb)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_5925_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: 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,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_5926_signed__and__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Va2: word(B),W: word(B)] : ring_1_signed(B,A,aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Va2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),ring_1_signed(B,A,Va2)),ring_1_signed(B,A,W)) ) ).

% signed_and_eq
tff(fact_5927_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_5928_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_5929_AND__upper2,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Y) ) ).

% AND_upper2
tff(fact_5930_AND__upper1,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Xa) ) ).

% AND_upper1
tff(fact_5931_AND__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)) ) ).

% AND_lower
tff(fact_5932_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_5933_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_5934_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_5935_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 = uminus_uminus(int,one_one(int)),
          L,
          $ite(L = 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),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% and_int_unfold
tff(fact_5936_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( aa(set(int),$o,member(int,K),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int)))))
        & aa(set(int),$o,member(int,L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,uminus_uminus(int,one_one(int))),bot_bot(set(int))))) ),
        uminus_uminus(int,
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
            & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,
          aa(int,fun(int,int),plus_plus(int),
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
              & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% and_int.simps
tff(fact_5937_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q4),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_5938_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_5939_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),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_ub(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_5940_word__and__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),uminus_uminus(word(A),one_one(word(A)))) = Xa ) ).

% word_and_max
tff(fact_5941_word__bitwise__m1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),uminus_uminus(word(A),one_one(word(A)))),Xa) = Xa ) ).

% word_bitwise_m1_simps(2)
tff(fact_5942_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_5943_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_5944_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_5945_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_5946_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_5947_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_5948_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_5949_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_5950_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_5951_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)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(6)
tff(fact_5952_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)),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(5)
tff(fact_5953_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)),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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(4)
tff(fact_5954_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),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_uc(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Nb)) ) ).

% divmod_algorithm_code(5)
tff(fact_5955_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))),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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_bitwise_1_simps(3)
tff(fact_5956_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)),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(5)
tff(fact_5957_word__and__le1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),A2)),A2) ) ).

% word_and_le1
tff(fact_5958_word__and__le2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),Y)),A2) ) ).

% word_and_le2
tff(fact_5959_word__log__esimps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_log_esimps(1)
tff(fact_5960_word__log__esimps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),zero_zero(word(A))),Xa) = zero_zero(word(A)) ) ).

% word_log_esimps(7)
tff(fact_5961_word__bw__assocs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: 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)),Xa),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) ) ).

% word_bw_assocs(1)
tff(fact_5962_word__bw__comms_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Xa) ) ).

% word_bw_comms(1)
tff(fact_5963_word__bw__same_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Xa) = Xa ) ).

% word_bw_same(1)
tff(fact_5964_word__bw__lcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) ) ).

% word_bw_lcs(1)
tff(fact_5965_word__wi__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),A2),B2)) ) ).

% word_wi_log_defs(2)
tff(fact_5966_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_ue(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_5967_nested__case__prod__simp,axiom,
    ! [D6: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D6,A))),Xa: product_prod(B,C),Y: D6] : aa(D6,A,aa(product_prod(B,C),fun(D6,A),aa(fun(B,fun(C,fun(D6,A))),fun(product_prod(B,C),fun(D6,A)),product_case_prod(B,C,fun(D6,A)),F),Xa),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D6,fun(B,fun(C,A)),aTP_Lamp_uf(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),F),Y)),Xa) ).

% nested_case_prod_simp
tff(fact_5968_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_ee(B,fun(A,product_prod(A,B)))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_5969_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,H: fun(B,fun(C,A)),F: fun(D6,B),Xs: list(D6),G: fun(D6,C)] : map(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H),zip(B,C,map(D6,B,F,Xs),map(D6,C,G,Xs))) = map(D6,A,aa(fun(D6,C),fun(D6,A),aa(fun(D6,B),fun(fun(D6,C),fun(D6,A)),aTP_Lamp_ug(fun(B,fun(C,A)),fun(fun(D6,B),fun(fun(D6,C),fun(D6,A))),H),F),G),Xs) ).

% map2_map_map
tff(fact_5970_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B,Xa: A,Xs: list(A)] : linorder_part(A,B,F,Pivot,aa(list(A),list(A),cons(A,Xa),Xs)) = aa(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A))),aa(fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),fun(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_ui(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),F),Pivot),Xa)),linorder_part(A,B,F,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_5971_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,map(C,A,F,Xs),Ys) = map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_uj(fun(C,A),fun(C,fun(B,product_prod(A,B))),F)),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_5972_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F: fun(C,B),Ys: list(C)] : zip(A,B,Xs,map(C,B,F,Ys)) = map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_uk(fun(C,B),fun(A,fun(C,product_prod(A,B))),F)),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_5973_map__prod__fun__zip,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,F: fun(C,A),G: fun(D6,B),Xs: list(C),Ys: list(D6)] : map(product_prod(C,D6),product_prod(A,B),aa(fun(C,fun(D6,product_prod(A,B))),fun(product_prod(C,D6),product_prod(A,B)),product_case_prod(C,D6,product_prod(A,B)),aa(fun(D6,B),fun(C,fun(D6,product_prod(A,B))),aTP_Lamp_ul(fun(C,A),fun(fun(D6,B),fun(C,fun(D6,product_prod(A,B)))),F),G)),zip(C,D6,Xs,Ys)) = zip(A,B,map(C,A,F,Xs),map(D6,B,G,Ys)) ).

% map_prod_fun_zip
tff(fact_5974_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D6: $tType,C: $tType,F: fun(product_prod(B,C),A),G: fun(D6,B),Xs: list(D6),Ys: list(C)] : map(product_prod(B,C),A,F,zip(B,C,map(D6,B,G,Xs),Ys)) = map(product_prod(D6,C),A,aa(fun(D6,fun(C,A)),fun(product_prod(D6,C),A),product_case_prod(D6,C,A),aa(fun(D6,B),fun(D6,fun(C,A)),aTP_Lamp_um(fun(product_prod(B,C),A),fun(fun(D6,B),fun(D6,fun(C,A))),F),G)),zip(D6,C,Xs,Ys)) ).

% map_zip_map
tff(fact_5975_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,F: fun(product_prod(B,C),A),Xs: list(B),G: fun(D6,C),Ys: list(D6)] : map(product_prod(B,C),A,F,zip(B,C,Xs,map(D6,C,G,Ys))) = map(product_prod(B,D6),A,aa(fun(B,fun(D6,A)),fun(product_prod(B,D6),A),product_case_prod(B,D6,A),aa(fun(D6,C),fun(B,fun(D6,A)),aTP_Lamp_un(fun(product_prod(B,C),A),fun(fun(D6,C),fun(B,fun(D6,A))),F),G)),zip(B,D6,Xs,Ys)) ).

% map_zip_map2
tff(fact_5976_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_5977_foldl__snd__zip,axiom,
    ! [B: $tType,C: $tType,A: $tType,Ys: list(A),Xs: list(B),F: fun(C,fun(A,C)),B2: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
     => ( foldl(C,product_prod(B,A),aTP_Lamp_up(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F),B2,zip(B,A,Xs,Ys)) = foldl(C,A,F,B2,Ys) ) ) ).

% foldl_snd_zip
tff(fact_5978_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_5979_and__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)
            & ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_5980_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),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_uq(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_5981_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),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ur(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_5982_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),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(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_us(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_5983_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),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_ut(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_5984_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set(product_prod(A,B)),F: fun(A,fun(B,C))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),A3)
     => aa(set(C),$o,member(C,aa(B,C,aa(A,fun(B,C),F,A2),B2)),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),A3)) ) ).

% pair_imageI
tff(fact_5985_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F,A2),B2) ).

% case_prod_conv
tff(fact_5986_case__prodI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F,A2),B2)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% case_prodI
tff(fact_5987_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A4: A,B3: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A4),B3) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P3) ) ).

% case_prodI2
tff(fact_5988_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B3: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => aa(set(C),$o,member(C,Z),aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B3)) )
     => aa(set(C),$o,member(C,Z),aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P3)) ) ).

% mem_case_prodI2
tff(fact_5989_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( aa(set(A),$o,member(A,Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,A2),B2))
     => aa(set(A),$o,member(A,Z),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2))) ) ).

% mem_case_prodI
tff(fact_5990_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),Xa: C] :
      ( ! [A4: A,B3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = P3 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A4),B3),Xa) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P3),Xa) ) ).

% case_prodI2'
tff(fact_5991_False__map2__and,axiom,
    ! [Xs: list($o),Ys: list($o)] :
      ( aa(set($o),$o,aa(set($o),fun(set($o),$o),ord_less_eq(set($o)),aa(list($o),set($o),set2($o),Xs)),aa(set($o),set($o),insert($o,$false),bot_bot(set($o))))
     => ( ( aa(list($o),nat,size_size(list($o)),Ys) = aa(list($o),nat,size_size(list($o)),Xs) )
       => ( map(product_prod($o,$o),$o,aa(fun($o,fun($o,$o)),fun(product_prod($o,$o),$o),product_case_prod($o,$o,$o),fconj),zip($o,$o,Xs,Ys)) = Xs ) ) ) ).

% False_map2_and
tff(fact_5992_False__map2__or,axiom,
    ! [Xs: list($o),Ys: list($o)] :
      ( aa(set($o),$o,aa(set($o),fun(set($o),$o),ord_less_eq(set($o)),aa(list($o),set($o),set2($o),Xs)),aa(set($o),set($o),insert($o,$false),bot_bot(set($o))))
     => ( ( aa(list($o),nat,size_size(list($o)),Ys) = aa(list($o),nat,size_size(list($o)),Xs) )
       => ( map(product_prod($o,$o),$o,aa(fun($o,fun($o,$o)),fun(product_prod($o,$o),$o),product_case_prod($o,$o,$o),fdisj),zip($o,$o,Xs,Ys)) = Ys ) ) ) ).

% False_map2_or
tff(fact_5993_UN__UN__split__split__eq,axiom,
    ! [D6: $tType,E4: $tType,A: $tType,C: $tType,B: $tType,A3: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Y6: set(product_prod(D6,E4)),X5: set(product_prod(B,C))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(set(product_prod(D6,E4)),fun(B,fun(C,set(A))),aTP_Lamp_uu(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(B,fun(C,set(A)))),A3),Y6))),X5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image(product_prod(B,C),set(A),aa(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A)),aTP_Lamp_ux(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A))),A3),Y6)),X5)) ).

% UN_UN_split_split_eq
tff(fact_5994_prod__case__refines,axiom,
    ! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),P5: product_prod(A,B),F: fun(A,fun(B,heap_Time_Heap(C))),F3: fun(A,fun(B,heap_Time_Heap(C)))] :
      ( ( P3 = P5 )
     => ( ! [A4: A,B3: B] : refine_Imp_refines(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F,A4),B3),aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F3,A4),B3))
       => refine_Imp_refines(C,aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),F),P3),aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),F3),P5)) ) ) ).

% prod_case_refines
tff(fact_5995_refines__case__prod__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,M: heap_Time_Heap(C),M7: fun(A,fun(B,heap_Time_Heap(C))),Ta: product_prod(A,B)] :
      ( ! [A4: A,B3: B] : refine_Imp_refines(C,M,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),M7,A4),B3))
     => refine_Imp_refines(C,M,aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),M7),Ta)) ) ).

% refines_case_prod_right
tff(fact_5996_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P3)
     => ~ ! [X: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X),Y3) ) ) ).

% case_prodE
tff(fact_5997_case__prodD,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))
     => aa(B,$o,aa(A,fun(B,$o),F,A2),B2) ) ).

% case_prodD
tff(fact_5998_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A2),B2),C2) ) ).

% case_prodD'
tff(fact_5999_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P3: product_prod(A,B),Z: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P3),Z)
     => ~ ! [X: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X),Y3),Z) ) ) ).

% case_prodE'
tff(fact_6000_Id__on__def_H,axiom,
    ! [A: $tType,A3: fun(A,$o)] : id_on(A,aa(fun(A,$o),set(A),collect(A),A3)) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_uy(fun(A,$o),fun(A,fun(A,$o)),A3))) ).

% Id_on_def'
tff(fact_6001_TBOUND__prod__case,axiom,
    ! [C: $tType,B: $tType,A: $tType,Ta: product_prod(A,B),F: fun(A,fun(B,heap_Time_Heap(C))),Bnd: fun(A,fun(B,nat))] :
      ( ! [A4: A,B3: B] :
          ( ( Ta = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F,A4),B3),aa(B,nat,aa(A,fun(B,nat),Bnd,A4),B3)) )
     => time_TBOUND(C,aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),F),Ta),aa(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),Bnd),Ta)) ) ).

% TBOUND_prod_case
tff(fact_6002_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A3: fun(A,fun(B,$o)),B4: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A3),B4)
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A3))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),B4))) ) ).

% Collect_case_prod_mono
tff(fact_6003_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uz(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_vb(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_6004_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xa: product_prod(A,B),P: assn,F: fun(A,fun(B,heap_Time_Heap(C))),Q: fun(C,assn)] :
      ( ! [A4: A,B3: B] :
          ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => hoare_hoare_triple(C,P,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F,A4),B3),Q) )
     => hoare_hoare_triple(C,P,aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),F),Xa),Q) ) ).

% case_prod_rule
tff(fact_6005_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_uz(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_vd(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_6006_align__lem__or,axiom,
    ! [Xs: list($o),Nb: nat,M: nat,Ys: list($o)] :
      ( ( aa(list($o),nat,size_size(list($o)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
     => ( ( aa(list($o),nat,size_size(list($o)),Ys) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
       => ( ( drop($o,M,Xs) = replicate($o,Nb,$false) )
         => ( ( take($o,M,Ys) = replicate($o,M,$false) )
           => ( map(product_prod($o,$o),$o,aa(fun($o,fun($o,$o)),fun(product_prod($o,$o),$o),product_case_prod($o,$o,$o),fdisj),zip($o,$o,Xs,Ys)) = append($o,take($o,M,Xs),drop($o,M,Ys)) ) ) ) ) ) ).

% align_lem_or
tff(fact_6007_align__lem__and,axiom,
    ! [Xs: list($o),Nb: nat,M: nat,Ys: list($o)] :
      ( ( aa(list($o),nat,size_size(list($o)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
     => ( ( aa(list($o),nat,size_size(list($o)),Ys) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
       => ( ( drop($o,M,Xs) = replicate($o,Nb,$false) )
         => ( ( take($o,M,Ys) = replicate($o,M,$false) )
           => ( map(product_prod($o,$o),$o,aa(fun($o,fun($o,$o)),fun(product_prod($o,$o),$o),product_case_prod($o,$o,$o),fconj),zip($o,$o,Xs,Ys)) = replicate($o,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M),$false) ) ) ) ) ) ).

% align_lem_and
tff(fact_6008_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ve(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_vb(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.triangle_reindex
tff(fact_6009_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ve(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_vd(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.triangle_reindex
tff(fact_6010_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_vf(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).

% zip_assoc
tff(fact_6011_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,$o))),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S3),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S3)))) ) ).

% Sup_SUP_eq2
tff(fact_6012_prod__encode__def,axiom,
    nat_prod_encode = aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),aTP_Lamp_vg(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_6013_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),$o)] : rel_of(A,B,M,P) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_vh(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P))) ).

% rel_of_def
tff(fact_6014_trancl__insert2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_vi(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R2)))) ).

% trancl_insert2
tff(fact_6015_foldr__snd__zip,axiom,
    ! [B: $tType,A: $tType,C: $tType,Ys: list(A),Xs: list(B),F: fun(A,fun(C,C)),B2: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
     => ( foldr(product_prod(B,A),C,aa(fun(B,fun(A,fun(C,C))),fun(product_prod(B,A),fun(C,C)),product_case_prod(B,A,fun(C,C)),aTP_Lamp_vj(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F)),zip(B,A,Xs,Ys),B2) = foldr(A,C,F,Ys,B2) ) ) ).

% foldr_snd_zip
tff(fact_6016_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D6: $tType,C: $tType,H: fun(B,A),F: fun(C,fun(D6,B)),Prod: product_prod(C,D6)] : aa(B,A,H,aa(product_prod(C,D6),B,aa(fun(C,fun(D6,B)),fun(product_prod(C,D6),B),product_case_prod(C,D6,B),F),Prod)) = aa(product_prod(C,D6),A,aa(fun(C,fun(D6,A)),fun(product_prod(C,D6),A),product_case_prod(C,D6,A),aa(fun(C,fun(D6,B)),fun(C,fun(D6,A)),aTP_Lamp_vk(fun(B,A),fun(fun(C,fun(D6,B)),fun(C,fun(D6,A))),H),F)),Prod) ).

% prod.case_distrib
tff(fact_6017_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),aTP_Lamp_vl(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_6018_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M,Ks,Vs) = foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_vn(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),M,zip(A,B,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_6019_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F,X),Y3) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) = G ) ) ).

% cond_case_prod_eta
tff(fact_6020_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_vo(fun(product_prod(A,B),C),fun(A,fun(B,C)),F)) = F ).

% case_prod_eta
tff(fact_6021_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( aa(A,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z))
     => ~ ! [X: B,Y3: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y3) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X),Y3)) ) ) ).

% case_prodE2
tff(fact_6022_quickcheck__narrowing__samples_Onarrowing__samples_Opsimps,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),I: code_integer,Zero: A] :
          ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),I)
         => ( code_T4080844693773952564amples(A,A_of_integer,Zero,I) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),I),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(A,fun(A,fun(A,list(A))),aa(code_integer,fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_vp(fun(code_integer,product_prod(A,A)),fun(code_integer,fun(A,fun(A,fun(A,list(A))))),A_of_integer),I),Zero)),aa(code_integer,product_prod(A,A),A_of_integer,I)),aa(list(A),list(A),cons(A,Zero),nil(A))) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.psimps
tff(fact_6023_quickcheck__narrowing__samples_Onarrowing__samples_Opelims,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,Xa: code_integer,Y: list(A)] :
          ( ( code_T4080844693773952564amples(A,A_of_integer,Zero,Xa) = Y )
         => ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),Xa)
           => ~ ( ( Y = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Xa),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_vq(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),Xa)),aa(code_integer,product_prod(A,A),A_of_integer,Xa)),aa(list(A),list(A),cons(A,Zero),nil(A))) )
               => ~ aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),Xa) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pelims
tff(fact_6024_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X4: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_vr($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X4)
    <=> ( (P)
        & aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q),X4) ) ) ).

% split_part
tff(fact_6025_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_vs(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_6026_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,Xa: code_integer,Y: list(A)] :
          ( ( code_T4080844693773952564amples(A,A_of_integer,Zero,Xa) = Y )
         => ( Y = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Xa),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_vq(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),Xa)),aa(code_integer,product_prod(A,A),A_of_integer,Xa)),aa(list(A),list(A),cons(A,Zero),nil(A))) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.elims
tff(fact_6027_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,I: code_integer] :
          code_T4080844693773952564amples(A,A_of_integer,Zero,I) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),I),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_vq(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),I)),aa(code_integer,product_prod(A,A),A_of_integer,I)),aa(list(A),list(A),cons(A,Zero),nil(A))) ) ).

% quickcheck_narrowing_samples.narrowing_samples.simps
tff(fact_6028_sort__key__by__quicksort__code,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F),Xs) = aa(list(A),list(A),case_list(list(A),A,nil(A),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_vx(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),F),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_6029_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,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_vy(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_6030_sort__upt,axiom,
    ! [M: nat,Nb: nat] : aa(list(nat),list(nat),linorder_sort_key(nat,nat,aTP_Lamp_kx(nat,nat)),upt(M,Nb)) = upt(M,Nb) ).

% sort_upt
tff(fact_6031_sort__upto,axiom,
    ! [I: int,J: int] : aa(list(int),list(int),linorder_sort_key(int,int,aTP_Lamp_ph(int,int)),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_6032_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),linorder_sort_key(A,B,F),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% length_sort
tff(fact_6033_sort__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_dw(A,A)) = linorder_quicksort(A) ) ) ).

% sort_quicksort
tff(fact_6034_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,aTP_Lamp_eo(B,fun(A,B),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_6035_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_nj(fun(A,B),fun(B,fun(A,$o)),F),K)),aa(list(A),list(A),linorder_sort_key(A,B,F),Xs)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_nj(fun(A,B),fun(B,fun(A,$o)),F),K)),Xs) ) ).

% sort_key_stable
tff(fact_6036_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_dw(A,A)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_6037_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_dw(A,A)),Xs)) ) ).

% sorted_sort
tff(fact_6038_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F,aa(list(B),list(B),linorder_sort_key(B,A,F),Xs))) ) ).

% sorted_sort_key
tff(fact_6039_sort__key__by__quicksort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F),Xs) = append(A,aa(list(A),list(A),linorder_sort_key(A,B,F),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_vz(fun(A,B),fun(list(A),fun(A,$o)),F),Xs)),Xs)),append(A,aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(list(A),fun(A,$o)),F),Xs)),Xs),aa(list(A),list(A),linorder_sort_key(A,B,F),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_wb(fun(A,B),fun(list(A),fun(A,$o)),F),Xs)),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_6040_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_dw(A,A)),Xs) = append(A,aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_dw(A,A)),aa(list(A),list(A),filter2(A,aTP_Lamp_wc(list(A),fun(A,$o),Xs)),Xs)),append(A,aa(list(A),list(A),filter2(A,aTP_Lamp_wd(list(A),fun(A,$o),Xs)),Xs),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_dw(A,A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Xs)))) ) ).

% sort_by_quicksort
tff(fact_6041_divmod__nat__if,axiom,
    ! [M: nat,Nb: nat] :
      divmod_nat(M,Nb) = $ite(
        ( ( Nb = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M),
        aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_we(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_6042_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),Nb)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),Nb))) ) ) ) ).

% and_mask_arith'
tff(fact_6043_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_wf(int,fun(int,fun(int,$o)),D))) ).

% int_ge_less_than_def
tff(fact_6044_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_6045_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_6046_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_6047_mask__eqs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),uminus_uminus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),uminus_uminus(word(A),A2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(10)
tff(fact_6048_mask__eqs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(9)
tff(fact_6049_mask__eqs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),Nb: nat,A2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb))),A2)),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B2),A2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(6)
tff(fact_6050_mask__eqs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(5)
tff(fact_6051_mask__eqs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(3)
tff(fact_6052_mask__eqs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(4)
tff(fact_6053_mask__eqs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(8)
tff(fact_6054_mask__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat,K: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),bit_se2239418461657761734s_mask(word(A),Nb))),K)),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),Xa),K)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_power_eq
tff(fact_6055_mask__eqs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(1)
tff(fact_6056_mask__eqs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(2)
tff(fact_6057_mask__eqs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(7)
tff(fact_6058_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),bit_se2239418461657761734s_mask(word(A),M))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ) ).

% mask_twice2
tff(fact_6059_le__mask__imp__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),bit_se2239418461657761734s_mask(word(A),Nb))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),bit_se2239418461657761734s_mask(word(A),Nb)) = Xa ) ) ) ).

% le_mask_imp_and_mask
tff(fact_6060_and__mask__eq__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) = W )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ).

% and_mask_eq_iff_le_mask
tff(fact_6061_mask__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,one_one(nat)) = one_one(A) ) ) ).

% mask_1
tff(fact_6062_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_6063_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_6064_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(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)),Xa),bit_se2239418461657761734s_mask(word(A),Nb)) = Xa ) ) ) ).

% less_mask_eq
tff(fact_6065_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_6066_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_6067_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Nb: nat,Va2: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),bit_se2239418461657761734s_mask(word(A),Nb)) = zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Va2),aa(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),Va2) = zero_zero(word(A)) ) ) ) ) ).

% is_aligned_AND_less_0
tff(fact_6068_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,size_size(word(A)),Xa))
         => aa(word(A),$o,aa(word(A),fun(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)),Xa),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_6069_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(word(A),fun(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_6070_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),one_one(word(A)))) ) ).

% word_and_mask_le_2pm1
tff(fact_6071_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),Xa,aa(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)),Xa),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_6072_and__mask__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),Nb)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),Nb))) ) ).

% and_mask_arith
tff(fact_6073_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_wg(int,fun(int,fun(int,$o)),D))) ).

% int_ge_less_than2_def
tff(fact_6074_word__2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,size_size(word(A)),W))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(int,fun(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_6075_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% mask_nat_positive_iff
tff(fact_6076_Word_Oof__nat__unat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [W: word(B)] : aa(nat,A,semiring_1_of_nat(A),aa(word(B),nat,semiring_1_unsigned(B,nat),W)) = aa(word(B),A,semiring_1_unsigned(B,A),W) ) ).

% Word.of_nat_unat
tff(fact_6077_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_6078_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_6079_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_nonnegative
tff(fact_6080_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),zero_zero(int))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),Xa) = zero_zero(int) ) ) ) ).

% uint_le_0_iff
tff(fact_6081_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)) ) ).

% uint_ge_0
tff(fact_6082_Word_Oof__int__uint,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [W: word(B)] : aa(int,A,ring_1_of_int(A),aa(word(B),int,semiring_1_unsigned(B,int),W)) = aa(word(B),A,semiring_1_unsigned(B,A),W) ) ).

% Word.of_int_uint
tff(fact_6083_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),zero_zero(int)) ) ).

% uint_lt_0
tff(fact_6084_word__le__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_le_no
tff(fact_6085_word__less__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(int,fun(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_6086_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_6087_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)))) ) ).

% word_div_no
tff(fact_6088_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_6089_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2239418461657761734s_mask(int,Nb)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_6090_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,Nb)) ).

% mask_nonnegative_int
tff(fact_6091_less__eq__mask,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ).

% less_eq_mask
tff(fact_6092_word__le__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) ) ).

% word_le_def
tff(fact_6093_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_greater_eq
tff(fact_6094_word__less__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(word(A),B,semiring_1_unsigned(A,B),A2)),aa(word(A),B,semiring_1_unsigned(A,B),B2)) ) ) ).

% word_less_eq_iff_unsigned
tff(fact_6095_word__less__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(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_6096_uint__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),Xa) = zero_zero(int) )
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% uint_0_iff
tff(fact_6097_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_6098_word__uint__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( A2 = B2 )
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),A2) = aa(word(A),int,semiring_1_unsigned(A,int),B2) ) ) ) ).

% word_uint_eq_iff
tff(fact_6099_word__uint__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),A2) = aa(word(A),int,semiring_1_unsigned(A,int),B2) )
         => ( A2 = B2 ) ) ) ).

% word_uint_eqI
tff(fact_6100_uint__cong,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
         => ( aa(word(A),int,semiring_1_unsigned(A,int),Xa) = aa(word(A),int,semiring_1_unsigned(A,int),Y) ) ) ) ).

% uint_cong
tff(fact_6101_word__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & semiring_char_0(B) )
     => ! [Va2: word(A),W: word(A)] :
          ( ( Va2 = W )
        <=> ( aa(word(A),B,semiring_1_unsigned(A,B),Va2) = aa(word(A),B,semiring_1_unsigned(A,B),W) ) ) ) ).

% word_eq_iff_unsigned
tff(fact_6102_unsigned__word__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [Va2: word(B),W: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),Va2) = aa(word(B),A,semiring_1_unsigned(B,A),W) )
         => ( Va2 = W ) ) ) ).

% unsigned_word_eqI
tff(fact_6103_word__of__int__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,word(A),ring_1_of_int(word(A)),aa(word(A),int,semiring_1_unsigned(A,int),W)) = W ) ).

% word_of_int_uint
tff(fact_6104_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_6105_uint__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Va2),W)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),Va2)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_div_distrib
tff(fact_6106_uint__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_div
tff(fact_6107_uint__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),modulo_modulo(word(A),Xa,Y)) = modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_mod
tff(fact_6108_uint__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),modulo_modulo(word(A),Va2,W)) = modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),Va2),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_mod_distrib
tff(fact_6109_uint__and,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_and
tff(fact_6110_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_6111_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(B,$o,aa(B,fun(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_6112_unsigned__and__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [Va2: word(B),W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Va2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(word(B),A,semiring_1_unsigned(B,A),Va2)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_and_eq
tff(fact_6113_udvd__iff__dvd__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] :
          ( udvd(A,Va2,W)
        <=> aa(int,$o,dvd_dvd(int,aa(word(A),int,semiring_1_unsigned(A,int),Va2)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ) ).

% udvd_iff_dvd_int
tff(fact_6114_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_add_ge0
tff(fact_6115_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_mult_ge0
tff(fact_6116_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: nat,Xa: word(A)] :
          ( ( nat2(aa(word(A),int,semiring_1_unsigned(A,int),Y)) = Z )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),nat2(aa(word(A),int,semiring_1_unsigned(A,int),Xa))),Z) ) ) ) ).

% nat_uint_less_helper
tff(fact_6117_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))) ) ).

% uint_add_le
tff(fact_6118_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple_iff
tff(fact_6119_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple
tff(fact_6120_no__ulen__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)),Xa)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)) ) ) ).

% no_ulen_sub
tff(fact_6121_word__add__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_add_def
tff(fact_6122_uint__sub__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xa),Y))) ) ).

% uint_sub_ge
tff(fact_6123_uint__sub__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xa),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_sub_lem
tff(fact_6124_word__minus__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : uminus_uminus(word(A),A2) = aa(int,word(A),ring_1_of_int(word(A)),uminus_uminus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2))) ) ).

% word_minus_def
tff(fact_6125_uint__minus__simple__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xa),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_minus_simple_alt
tff(fact_6126_uint__minus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)),Xa)
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xa),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_minus_simple_iff
tff(fact_6127_less__mask,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ) ).

% less_mask
tff(fact_6128_word__sub__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),minus_minus(word(A),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_sub_wi
tff(fact_6129_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),A2),Nb) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),Nb)) ) ).

% word_arith_power_alt
tff(fact_6130_word__mult__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_mult_def
tff(fact_6131_word__and__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_and_def
tff(fact_6132_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_div_def
tff(fact_6133_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_6134_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Nb: int,K4: word(A),N3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K4))),Uq) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_6135_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,Nb: int,K4: word(A),N3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K4))),Uq) ) ) ) ) ).

% udvd_incr_lem
tff(fact_6136_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q4: word(A),Nb: int,K4: word(A),N3: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q4)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q4) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4)),Q4) ) ) ) ) ).

% udvd_incr0
tff(fact_6137_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q4: word(A),Nb: int,K4: word(A),N3: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q4)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q4) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q4) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),minus_minus(word(A),Q4),K4)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_6138_udvd__unfold__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( udvd(A,A2,B2)
        <=> ? [N6: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N6)
              & ( aa(word(A),int,semiring_1_unsigned(A,int),B2) = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),A2)) ) ) ) ) ).

% udvd_unfold_int
tff(fact_6139_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q4: word(A),Ua: int,Nb: int,K4: word(A),N3: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q4)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q4) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4)),Q4) ) ) ) ) ).

% udvd_incr'
tff(fact_6140_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q4: word(A),Ua: int,Nb: int,K4: word(A),N3: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q4)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q4) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q4) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),minus_minus(word(A),Q4),K4)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_6141_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_6142_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$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_6143_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,aa(int,fun(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_6144_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(int,$o,aa(int,fun(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_6145_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W))
          & aa(int,$o,aa(int,fun(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_6146_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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_6147_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_6148_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_6149_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(int,$o,aa(int,fun(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),Xa)),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)),Xa))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_6150_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),Xa),Y)) = $ite(aa(int,$o,aa(int,fun(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),Xa)),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)),Xa))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),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),Xa)),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)),Xa)))) ) ).

% uint_plus_if_size
tff(fact_6151_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xa),Y)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),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),Xa)),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)),Xa)))) ) ).

% uint_sub_if_size
tff(fact_6152_arctan__def,axiom,
    ! [Y: real] : arctan(Y) = the(real,aTP_Lamp_wh(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_6153_arcsin__def,axiom,
    ! [Y: real] : arcsin(Y) = the(real,aTP_Lamp_wi(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_6154_udvdI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(A),Va2: word(A),U: word(B)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),W) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Va2)),aa(word(B),nat,semiring_1_unsigned(B,nat),U)) )
         => udvd(A,Va2,W) ) ) ).

% udvdI
tff(fact_6155_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != 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),Xa),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% Suc_unat_minus_one
tff(fact_6156_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% word_le_nat_alt
tff(fact_6157_le__ucast__ucast__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Y: word(B)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),Y) ) ) ).

% le_ucast_ucast_le
tff(fact_6158_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: nat,Z: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(word(A),nat,semiring_1_unsigned(A,nat),Z))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)) = Y ) ) ) ).

% le_unat_uoi
tff(fact_6159_uno__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Nb: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),Nb) ) ).

% uno_simps(2)
tff(fact_6160_unat__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Va2),W)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Va2)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_div_distrib
tff(fact_6161_unat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_div
tff(fact_6162_max__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_max(word(A)),A2),B2)),C2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_max(word(A)),A2),B2))),aa(word(A),nat,semiring_1_unsigned(A,nat),C2)) ) ).

% max_lt
tff(fact_6163_unat__eq__nat__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),W) = nat2(aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% unat_eq_nat_uint
tff(fact_6164_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_6165_ucast__0__I,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A)] :
          ( ( Xa = zero_zero(word(A)) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa) = zero_zero(word(B)) ) ) ) ).

% ucast_0_I
tff(fact_6166_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_6167_word__unat__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] :
          ( ( Va2 = W )
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),Va2) = aa(word(A),nat,semiring_1_unsigned(A,nat),W) ) ) ) ).

% word_unat_eq_iff
tff(fact_6168_unat__cong,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = aa(word(A),nat,semiring_1_unsigned(A,nat),Y) ) ) ) ).

% unat_cong
tff(fact_6169_uint__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),W) = aa(nat,int,semiring_1_of_nat(int),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% uint_nat
tff(fact_6170_ucast__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),aa(word(B),int,semiring_1_unsigned(B,int),W)) ) ).

% ucast_eq
tff(fact_6171_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_6172_unat__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
         => aa(nat,$o,aa(nat,fun(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_6173_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(nat,$o,aa(nat,fun(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_6174_unat__eq__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = zero_zero(nat) )
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% unat_eq_zero
tff(fact_6175_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_6176_unat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),modulo_modulo(word(A),Xa,Y)) = modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_mod
tff(fact_6177_unat__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),modulo_modulo(word(A),Va2,W)) = modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Va2),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_mod_distrib
tff(fact_6178_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat,Y: word(A)] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),Nb)
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),Nb) )
         => aa(nat,$o,aa(nat,fun(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)),Xa),Y))),Nb) ) ) ).

% word_unat_and_lt
tff(fact_6179_udvd__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( udvd(A,A2,B2)
        <=> ? [N6: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),B2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N6),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)) ) ) ).

% udvd_nat_alt
tff(fact_6180_udvdE,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Va2: word(A),W: word(A)] :
          ( udvd(A,Va2,W)
         => ~ ! [U4: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),W) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Va2)),aa(word(A),nat,semiring_1_unsigned(A,nat),U4)) ) ) ).

% udvdE
tff(fact_6181_udvd__iff__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( udvd(A,Xa,Y)
        <=> aa(nat,$o,dvd_dvd(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ).

% udvd_iff_dvd
tff(fact_6182_ln__real__def,axiom,
    ! [Xa: real] : ln_ln(real,Xa) = the(real,aTP_Lamp_wj(real,fun(real,$o),Xa)) ).

% ln_real_def
tff(fact_6183_suminf__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F: fun(nat,A)] : suminf(A,F) = the(A,sums(A,F)) ) ).

% suminf_def
tff(fact_6184_unat__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = aa(nat,nat,suc,zero_zero(nat)) )
        <=> ( Xa = one_one(word(A)) ) ) ) ).

% unat_eq_1
tff(fact_6185_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ( Xa != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_6186_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(B),nat,semiring_1_unsigned(B,nat),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(B),int,semiring_1_unsigned(B,int),B2)) ) ) ).

% un_ui_le
tff(fact_6187_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),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)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_plus_simple
tff(fact_6188_unat__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),A2)
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ) ).

% unat_sub
tff(fact_6189_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),Nb) ) ) ).

% unat_less_helper
tff(fact_6190_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)),Xa) ) ) ).

% word_of_nat_less
tff(fact_6191_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),B2) ) ) ).

% word_unat_less_le
tff(fact_6192_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)),Xa) ) ) ).

% word_of_nat_le
tff(fact_6193_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_6194_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_6195_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_div
tff(fact_6196_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_6197_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( ln_ln(real,Xa) = the(real,aTP_Lamp_wk(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_6198_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)) ) ) ).

% unat_1_0
tff(fact_6199_unat__max__word__pos,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),uminus_uminus(word(A),one_one(word(A))))) ) ).

% unat_max_word_pos
tff(fact_6200_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_6201_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_6202_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xa),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% Suc_unat_diff_1
tff(fact_6203_unat__Suc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] :
          ( ( Nb != 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_6204_uno__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),M: nat] :
          ( aa(nat,$o,aa(nat,fun(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_6205_measure__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A)] :
          ( ( P3 != zero_zero(word(A)) )
         => aa(nat,$o,aa(nat,fun(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_6206_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_wl(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_6207_word__overflow__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),Xa),one_one(word(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),one_one(nat)) )
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow_unat
tff(fact_6208_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_6209_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,MaxBound: word(A),Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_6210_even__word__imp__odd__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),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)),Xa),one_one(word(A))))) ) ) ) ).

% even_word_imp_odd_next
tff(fact_6211_odd__word__imp__even__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),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)),Xa),one_one(word(A))))) ) ) ) ).

% odd_word_imp_even_next
tff(fact_6212_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Nb),M) = one_one(word(A)) )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),M),Nb)
            & aa(nat,$o,aa(nat,fun(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_6213_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_6214_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 )
        <=> ? [Q6: 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),Q6),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_6215_pi__half,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) = the(real,aTP_Lamp_wm(real,$o)) ).

% pi_half
tff(fact_6216_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_wm(real,$o))) ).

% pi_def
tff(fact_6217_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(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),Xa)),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)),Xa))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),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),Xa)),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)),Xa)))) ) ).

% unat_plus_if_size
tff(fact_6218_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),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),Xa)),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)),Xa)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if_size
tff(fact_6219_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(nat,$o,aa(nat,fun(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),Xa)),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)),Xa))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_6220_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(word(A),nat,size_size(word(A)),W))
         => aa(nat,$o,aa(nat,fun(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_6221_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ( X = A2 ) )
       => ( the(A,P) = A2 ) ) ) ).

% the_equality
tff(fact_6222_the__eq__trivial,axiom,
    ! [A: $tType,A2: A] : the(A,aTP_Lamp_ca(A,fun(A,$o),A2)) = A2 ).

% the_eq_trivial
tff(fact_6223_ucast__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),semiring_1_unsigned(A,word(A)),W) = W ) ).

% ucast_id
tff(fact_6224_the__sym__eq__trivial,axiom,
    ! [A: $tType,Xa: A] : the(A,aa(A,fun(A,$o),fequal(A),Xa)) = Xa ).

% the_sym_eq_trivial
tff(fact_6225_The__split__eq,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_wn(A,fun(B,fun(A,fun(B,$o))),Xa),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) ).

% The_split_eq
tff(fact_6226_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X4 ) ) )
     => ( aa(A,$o,P,A2)
       => ( the(A,P) = A2 ) ) ) ).

% the1_equality
tff(fact_6227_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X4 ) ) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => aa(A,$o,Q,X) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_6228_If__def,axiom,
    ! [A: $tType,P: $o,Xa: A,Y: A] :
      $ite((P),Xa,Y) = the(A,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_wo($o,fun(A,fun(A,fun(A,$o))),(P)),Xa),Y)) ).

% If_def
tff(fact_6229_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ( X = A2 ) )
       => ( ! [X: A] :
              ( aa(A,$o,P,X)
             => aa(A,$o,Q,X) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_6230_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X4 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_6231_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ( X = A2 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_6232_floor__real__def,axiom,
    ! [Xa: real] : archim6421214686448440834_floor(real,Xa) = the(int,aTP_Lamp_wp(real,fun(int,$o),Xa)) ).

% floor_real_def
tff(fact_6233_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,size_size(word(A)),W))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)) ) ) ) ).

% neg_mask_is_div'
tff(fact_6234_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_6235_word__not__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)) = Xa ) ).

% word_not_not
tff(fact_6236_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_6237_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_6238_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_6239_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_6240_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),Xa) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_6241_word__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)) = zero_zero(word(A)) ) ).

% word_and_not
tff(fact_6242_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_6243_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_6244_bit_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),zero_zero(A)) = uminus_uminus(A,one_one(A)) ) ) ).

% bit.compl_zero
tff(fact_6245_bit_Ocompl__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),uminus_uminus(A,one_one(A))) = zero_zero(A) ) ) ).

% bit.compl_one
tff(fact_6246_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_6247_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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ).

% take_bit_of_Suc_0
tff(fact_6248_word__bitwise__m1__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),uminus_uminus(word(A),one_one(word(A)))) = zero_zero(word(A)) ) ) ).

% word_bitwise_m1_simps(1)
tff(fact_6249_word__add__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)) = uminus_uminus(word(A),one_one(word(A))) ) ).

% word_add_not
tff(fact_6250_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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% take_bit_of_1
tff(fact_6251_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_6252_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_6253_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,$o,aa(nat,fun(nat,$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_6254_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_6255_uint__take__bit__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),bit_se2584673776208193580ke_bit(word(A),Nb),W)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_take_bit_eq
tff(fact_6256_unsigned__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [Nb: nat,W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),bit_se2584673776208193580ke_bit(word(B),Nb),W)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_take_bit_eq
tff(fact_6257_mask__eq__0__eq__x,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),W) = zero_zero(word(A)) )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = Xa ) ) ) ).

% mask_eq_0_eq_x
tff(fact_6258_mask__eq__x__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),W) = Xa )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = zero_zero(word(A)) ) ) ) ).

% mask_eq_x_eq_0
tff(fact_6259_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_6260_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_6261_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_6262_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Nb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_6263_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_nonnegative
tff(fact_6264_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_6265_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Nb: nat,Q4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q4)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Q4)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_6266_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_6267_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_6268_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),nat2(K)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_6269_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),nat2(K)) ) ) ).

% nat_take_bit_eq
tff(fact_6270_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),bit_se2584673776208193580ke_bit(A,Nb),bit_ri4674362597316999326ke_bit(A,M)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_6271_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se8732182000553998342ip_bit(A,M,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_6272_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se5668285175392031749et_bit(A,M,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se5668285175392031749et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_6273_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se2638667681897837118et_bit(A,M,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se2638667681897837118et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_6274_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_6275_less__eq__mask__iff__take__bit__eq__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))
        <=> ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Nb),W) = W ) ) ) ).

% less_eq_mask_iff_take_bit_eq_self
tff(fact_6276_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_6277_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_lower_twice
tff(fact_6278_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat,Y: word(A),M: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) = Y )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ) ).

% mask_out_first_mask_some
tff(fact_6279_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_6280_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_6281_and__mask__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),ring_1_of_int(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),I)) ) ).

% and_mask_wi
tff(fact_6282_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% multiple_mask_trivia
tff(fact_6283_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) = zero_zero(word(A)) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_6284_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),uminus_uminus(int,one_one(int)))) ) ).

% mask_bin
tff(fact_6285_and__mask__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(word(A),int,semiring_1_unsigned(A,int),W))) ) ).

% and_mask_bintr
tff(fact_6286_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_6287_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_6288_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M) = M )
    <=> aa(nat,$o,aa(nat,fun(nat,$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_6289_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$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_6290_take__bit__nat__eq__self,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_6291_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_6292_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_6293_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,$o,aa(nat,fun(nat,$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_6294_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M)),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_6295_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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),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_6296_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$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_6297_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_6298_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$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_6299_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$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_6300_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_6301_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)),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_6302_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_6303_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6304_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,minus_minus(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_6305_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_6306_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_6307_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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_6308_floor__rat__def,axiom,
    ! [Xa: rat] : archim6421214686448440834_floor(rat,Xa) = the(int,aTP_Lamp_wq(rat,fun(int,$o),Xa)) ).

% floor_rat_def
tff(fact_6309_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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),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_6310_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_6311_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_6312_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_6313_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),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_6314_word__no__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2)))) ) ).

% word_no_log_defs(2)
tff(fact_6315_word__wi__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(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,bit_ri4277139882892585799ns_not(int),A2)) ) ).

% word_wi_log_defs(1)
tff(fact_6316_less__eq__rat__def,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xa),Y)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_rat_def
tff(fact_6317_abs__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,abs_abs(rat),A2) = $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),A2),zero_zero(rat)),uminus_uminus(rat,A2),A2) ).

% abs_rat_def
tff(fact_6318_sgn__rat__def,axiom,
    ! [A2: rat] :
      sgn_sgn(rat,A2) = $ite(
        A2 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A2),one_one(rat),uminus_uminus(rat,one_one(rat))) ) ).

% sgn_rat_def
tff(fact_6319_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
     => ~ ! [S: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S)
           => ! [T6: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T6)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_6320_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_6321_word__not__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),A2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(word(A),int,semiring_1_unsigned(A,int),A2))) ) ).

% word_not_def
tff(fact_6322_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_6323_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)),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)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),inc(M)))))) ) ).

% take_bit_word_minus_Bit1_eq
tff(fact_6324_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_6325_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_6326_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_6327_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_6328_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Nb) ) ).

% less_numeral_Suc
tff(fact_6329_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_6330_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_6331_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_6332_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_6333_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_6334_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_6335_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Nb)) ).

% max_numeral_Suc
tff(fact_6336_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_6337_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)),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)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_minus_Bit0_eq
tff(fact_6338_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_6339_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_6340_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),pred_numeral(K)),aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_6341_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_6342_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_6343_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_6344_int__not__code_I1_J,axiom,
    aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) = uminus_uminus(int,one_one(int)) ).

% int_not_code(1)
tff(fact_6345_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = uminus_uminus(int,one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = uminus_uminus(int,one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,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),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_6346_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),Xa) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_6347_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_6348_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_6349_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_6350_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_6351_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_6352_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_6353_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: 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,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_6354_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: 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,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_6355_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_6356_XOR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Y)) ) ) ).

% XOR_lower
tff(fact_6357_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_6358_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_6359_signed__xor__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Va2: word(B),W: word(B)] : ring_1_signed(B,A,aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344971392196577ns_xor(word(B)),Va2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),ring_1_signed(B,A,Va2)),ring_1_signed(B,A,W)) ) ).

% signed_xor_eq
tff(fact_6360_unsigned__xor__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [Va2: word(B),W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344971392196577ns_xor(word(B)),Va2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(word(B),A,semiring_1_unsigned(B,A),Va2)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_xor_eq
tff(fact_6361_XOR__upper,axiom,
    ! [Xa: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),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_6362_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_6363_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_6364_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_6365_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( arcosh(real,Xa) = ln_ln(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),sqrt(aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_6366_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_6367_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xa: real] :
      ( ( sqrt(Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_6368_real__sqrt__zero,axiom,
    sqrt(zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_6369_real__sqrt__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(Xa)),sqrt(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% real_sqrt_le_iff
tff(fact_6370_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sqrt(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_6371_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sqrt(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_6372_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sqrt(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_6373_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_6374_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),sqrt(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_6375_real__sqrt__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_6376_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_6377_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_6378_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_6379_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$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_6380_word__bitwise__m1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),uminus_uminus(word(A),one_one(word(A)))) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa) ) ).

% word_bitwise_m1_simps(7)
tff(fact_6381_word__bitwise__m1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),uminus_uminus(word(A),one_one(word(A)))),Xa) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa) ) ).

% word_bitwise_m1_simps(6)
tff(fact_6382_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xa)) ).

% xor_nat_numerals(4)
tff(fact_6383_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa)) ).

% xor_nat_numerals(3)
tff(fact_6384_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_6385_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_6386_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,uminus_uminus(int,aa(num,int,numeral_numeral(int),bit0(W)))),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,uminus_uminus(int,aa(num,int,numeral_numeral(int),W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_6387_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,uminus_uminus(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(nat,nat,suc,Nb))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_6388_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_6389_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_6390_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_6391_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),sqrt(Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_6392_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),sqrt(Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_6393_bin__nth__minus__Bit0,axiom,
    ! [Nb: nat,W: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_6394_bin__nth__minus__Bit1,axiom,
    ! [Nb: nat,W: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_6395_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_6396_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_6397_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)),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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(12)
tff(fact_6398_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)),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(13)
tff(fact_6399_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)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(14)
tff(fact_6400_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))),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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_bitwise_1_simps(11)
tff(fact_6401_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)),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(13)
tff(fact_6402_word__bw__same_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Xa) = zero_zero(word(A)) ) ).

% word_bw_same(3)
tff(fact_6403_word__log__esimps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),zero_zero(word(A))),Xa) = Xa ) ).

% word_log_esimps(11)
tff(fact_6404_word__log__esimps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),zero_zero(word(A))) = Xa ) ).

% word_log_esimps(5)
tff(fact_6405_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_6406_word__bw__assocs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: 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)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),Z)) ) ).

% word_bw_assocs(3)
tff(fact_6407_word__bw__comms_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),Xa) ) ).

% word_bw_comms(3)
tff(fact_6408_word__bw__lcs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: 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_se5824344971392196577ns_xor(word(A)),Xa),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),Z)) ) ).

% word_bw_lcs(3)
tff(fact_6409_uint__xor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_xor
tff(fact_6410_word__wi__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),A2),B2)) ) ).

% word_wi_log_defs(4)
tff(fact_6411_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_take_bit_iff
tff(fact_6412_real__sqrt__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(Xa)),sqrt(Y)) ) ).

% real_sqrt_le_mono
tff(fact_6413_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_6414_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_6415_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_6416_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sqrt(Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_6417_real__sqrt__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),sqrt(Xa)) ) ).

% real_sqrt_ge_one
tff(fact_6418_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sqrt(Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_6419_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( sqrt(Xa) = zero_zero(real) )
       => ( Xa = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_6420_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),sqrt(Xa)) = sqrt(Xa) ) ) ).

% real_div_sqrt
tff(fact_6421_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),sqrt(Xa)),sqrt(Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_6422_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_6423_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) )
     => ( map(nat,$o,bit_se5641148757651400278ts_bit(int,K),upt(zero_zero(nat),Nb)) = map(nat,$o,bit_se5641148757651400278ts_bit(int,L),upt(zero_zero(nat),Nb)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_6424_word__xor__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_xor_def
tff(fact_6425_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),sqrt(Xa)),Xa) = aa(real,real,inverse_inverse(real),sqrt(Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_6426_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_6427_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_6428_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb) ) ) ).

% bit_Suc
tff(fact_6429_sqrt__le__D,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(Xa)),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),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_6430_real__le__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),sqrt(Y)) ) ).

% real_le_rsqrt
tff(fact_6431_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$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_6432_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_6433_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_6434_real__sqrt__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( sqrt(Xa) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_6435_real__le__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(Xa)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_6436_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),U)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),sqrt(aa(num,real,numeral_numeral(real),bit0(one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_6437_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xa: real,Y: real] :
      ( ( sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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 )
     => ( Xa = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_6438_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( ( sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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))))) = Xa )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_6439_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),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)))))),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_6440_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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_6441_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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_6442_sqrt__ge__absD,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),sqrt(Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_6443_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_6444_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_6445_real__less__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sqrt(Xa)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_6446_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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),Xa),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_6447_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),sqrt(Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_6448_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_6449_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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_6450_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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_6451_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ln_ln(real,sqrt(Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),ln_ln(real,Xa)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_6452_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),
            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)),Xa),Y)),
              $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),uminus_uminus(word(A),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_max(word(A)),Xa),Y) ) ).

% bit_twiddle_max
tff(fact_6453_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(list($o),A,groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2))),Bs)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(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_6454_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2))
         => ( 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_6455_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_6456_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),sqrt(Xa)),Nb) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_6457_arsinh__real__aux,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_6458_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(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),Xa),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_6459_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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_6460_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sqrt(aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_6461_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = sqrt(Xa) ) ) ).

% powr_half_sqrt
tff(fact_6462_xor__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_6463_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : aa(list($o),A,groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2))),map(nat,$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_6464_cos__x__y__le__one,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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_6465_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_6466_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_6467_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_6468_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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_6469_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xa))
     => ( sin(real,Xa) = sqrt(aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_6470_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( cos(real,arcsin(Xa)) = sqrt(aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_6471_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( sin(real,arccos(Y)) = 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_6472_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( sin(real,arccos(Xa)) = sqrt(aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_6473_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_6474_arccos__1,axiom,
    arccos(one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_6475_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( cos(real,arccos(Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_6476_map__nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(nat)] : map(nat,$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_6477_arccos__0,axiom,
    arccos(zero_zero(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arccos_0
tff(fact_6478_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Xa)),Nb)
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb) ) ) ).

% test_bit_over
tff(fact_6479_word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Va2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(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),Va2),N) ) )
         => ( U = Va2 ) ) ) ).

% word_eqI
tff(fact_6480_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,size_size(word(A)),W)) ) ) ).

% test_bit_size
tff(fact_6481_finite__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),bit_se5641148757651400278ts_bit(word(A),W))) ) ).

% finite_bit_word
tff(fact_6482_word__exists__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ? [X_13: nat] : aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),X_13) ) ) ).

% word_exists_nth
tff(fact_6483_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_6484_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_6485_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_6486_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_6487_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),zero_zero(nat))
         => aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),zero_zero(nat)) ) ) ).

% lsb_this_or_next
tff(fact_6488_word__leI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Va2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(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,$o,bit_se5641148757651400278ts_bit(word(A),Va2),N) ) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),U),Va2) ) ) ).

% word_leI
tff(fact_6489_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,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(word(A),nat,size_size(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) ) ) ) ).

% nth_mask
tff(fact_6490_arccos__le__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),arccos(Xa)) ) ) ) ).

% arccos_le_arccos
tff(fact_6491_arccos__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Xa)),arccos(Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ).

% arccos_le_mono
tff(fact_6492_arccos__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)) )
     => ( ( arccos(Xa) = arccos(Y) )
      <=> ( Xa = Y ) ) ) ).

% arccos_eq_iff
tff(fact_6493_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,nat2(K)),Nb)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ).

% bit_nat_iff
tff(fact_6494_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) )
         => aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),zero_zero(nat)) ) ) ).

% overflow_imp_lsb
tff(fact_6495_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_6496_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,$o,aa(nat,fun(nat,$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_6497_le__mask__high__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))
        <=> ! [X3: nat] :
              ( aa(set(nat),$o,member(nat,X3),set_or7035219750837199246ssThan(nat,Nb,aa(word(A),nat,size_size(word(A)),W)))
             => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),X3) ) ) ) ).

% le_mask_high_bits
tff(fact_6498_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arccos(Y)) ) ) ).

% arccos_lbound
tff(fact_6499_arccos__less__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),arccos(Y)),arccos(Xa)) ) ) ) ).

% arccos_less_arccos
tff(fact_6500_arccos__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arccos(Xa)),arccos(Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ).

% arccos_less_mono
tff(fact_6501_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),pi) ) ) ).

% arccos_ubound
tff(fact_6502_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( arccos(cos(real,Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_6503_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( cos(real,arccos(Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_6504_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi)
     => ( arccos(cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_6505_bang__is__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),M)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),Xa) ) ) ).

% bang_is_le
tff(fact_6506_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),arccos(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arccos(Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_6507_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arccos(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_6508_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),zero_zero(nat)) ) ) ).

% odd_iff_lsb
tff(fact_6509_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( sin(real,arccos(Xa)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_6510_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,pi)),Xa)
       => ( arccos(cos(real,Xa)) = uminus_uminus(real,Xa) ) ) ) ).

% arccos_cos2
tff(fact_6511_arccos__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( arccos(uminus_uminus(real,Xa)) = aa(real,real,minus_minus(real,pi),arccos(Xa)) ) ) ) ).

% arccos_minus
tff(fact_6512_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(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_6513_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: nat,Xa: word(A)] :
          ( ( Y = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y) != zero_zero(word(A)) )
          <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb) ) ) ) ).

% and_neq_0_is_nth
tff(fact_6514_arccos__def,axiom,
    ! [Y: real] : arccos(Y) = the(real,aTP_Lamp_wr(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_6515_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arccos(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),pi)
          & ( cos(real,arccos(Y)) = Y ) ) ) ) ).

% arccos
tff(fact_6516_arccos__minus__abs,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( arccos(uminus_uminus(real,Xa)) = aa(real,real,minus_minus(real,pi),arccos(Xa)) ) ) ).

% arccos_minus_abs
tff(fact_6517_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_6518_test__bit__rcat,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Sw: nat,Wl: list(word(A)),Rc: word(B),Nb: nat] :
          ( ( Sw = aa(word(A),nat,size_size(word(A)),aa(list(word(A)),word(A),hd(word(A)),Wl)) )
         => ( ( Rc = aa(list(word(A)),word(B),word_rcat(A,B),Wl) )
           => ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(B),Rc),Nb)
            <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(B),nat,size_size(word(B)),Rc))
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Sw)),aa(list(word(A)),nat,size_size(list(word(A))),Wl))
                & aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(nat,word(A),nth(word(A),aa(list(word(A)),list(word(A)),rev(word(A)),Wl)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Sw))),modulo_modulo(nat,Nb,Sw)) ) ) ) ) ) ).

% test_bit_rcat
tff(fact_6519_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_6520_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_6521_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_6522_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = uminus_uminus(int,one_one(int)) )
        | ( L = uminus_uminus(int,one_one(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,aa(int,fun(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),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% or_int_unfold
tff(fact_6523_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,$o),Xa: A,Xs: list(A)] :
      extract(A,P,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(aa(A,$o,P,Xa),aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Xa),Xs))),case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_wt(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Xa)),extract(A,P,Xs))) ).

% extract_Cons_code
tff(fact_6524_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_6525_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_6526_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_6527_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_6528_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_6529_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: 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,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_6530_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: 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,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_6531_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_6532_OR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Y)) ) ) ).

% OR_lower
tff(fact_6533_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_6534_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_6535_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),zero_zero(A)) = Xa ) ).

% bit.disj_zero_right
tff(fact_6536_option_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X2: B] : case_option(A,B,F1,F22,aa(B,option(B),some(B),X2)) = aa(B,A,F22,X2) ).

% option.simps(5)
tff(fact_6537_option_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,A)] : case_option(A,B,F1,F22,none(B)) = F1 ).

% option.simps(4)
tff(fact_6538_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,B),Option: option(C)] : aa(B,A,H,case_option(B,C,F1,F22,Option)) = case_option(A,C,aa(B,A,H,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_ea(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22),Option) ).

% option.case_distrib
tff(fact_6539_signed__or__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Va2: word(B),W: word(B)] : ring_1_signed(B,A,aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se1065995026697491101ons_or(word(B)),Va2),W)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),ring_1_signed(B,A,Va2)),ring_1_signed(B,A,W)) ) ).

% signed_or_eq
tff(fact_6540_unsigned__or__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [Va2: word(B),W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se1065995026697491101ons_or(word(B)),Va2),W)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(word(B),A,semiring_1_unsigned(B,A),Va2)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_or_eq
tff(fact_6541_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_6542_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_6543_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_6544_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),Option: option(B)] :
      case_option(A,B,F1,F22,Option) = $ite(Option = none(B),F1,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ).

% option.case_eq_if
tff(fact_6545_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xa) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xa) = 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) = uminus_uminus(A,one_one(A)) )
               => ( Xa = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_6546_option_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F22,Option))
    <=> ( ( ( Option = none(B) )
         => aa(A,$o,P,F1) )
        & ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
         => aa(A,$o,P,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel
tff(fact_6547_option_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F22,Option))
    <=> ~ ( ( ( Option = none(B) )
            & ~ aa(A,$o,P,F1) )
          | ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
            & ~ aa(A,$o,P,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_6548_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),Y) = uminus_uminus(A,one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_6549_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_6550_sup__option__def,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Xa: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),Xa),Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_wv(option(A),fun(option(A),fun(A,option(A))),Xa),Y),Xa) ) ).

% sup_option_def
tff(fact_6551_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_6552_inf__option__def,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Xa: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),Xa),Y) = case_option(option(A),A,none(A),aTP_Lamp_wx(option(A),fun(A,option(A)),Y),Xa) ) ).

% inf_option_def
tff(fact_6553_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_6554_OR__upper,axiom,
    ! [Xa: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),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_6555_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_6556_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_wy(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_6557_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),uminus_uminus(A,one_one(A)),bit_se5824344971392196577ns_xor(A)) ) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_6558_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)),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(9)
tff(fact_6559_word__ao__absorbs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: 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),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),Xa) = Xa ) ).

% word_ao_absorbs(8)
tff(fact_6560_word__ao__absorbs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: 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_se1065995026697491101ons_or(word(A)),Xa),Y)),Xa) = Xa ) ).

% word_ao_absorbs(7)
tff(fact_6561_word__ao__absorbs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)) = Xa ) ).

% word_ao_absorbs(6)
tff(fact_6562_word__ao__absorbs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: 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_se1065995026697491101ons_or(word(A)),Y),Xa)),Xa) = Xa ) ).

% word_ao_absorbs(5)
tff(fact_6563_word__ao__absorbs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: 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),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Xa)),Xa) = Xa ) ).

% word_ao_absorbs(4)
tff(fact_6564_word__ao__absorbs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) = Xa ) ).

% word_ao_absorbs(3)
tff(fact_6565_word__ao__absorbs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Xa)) = Xa ) ).

% word_ao_absorbs(2)
tff(fact_6566_word__ao__absorbs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Xa)) = Xa ) ).

% word_ao_absorbs(1)
tff(fact_6567_word__plus__and__or,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y) ) ).

% word_plus_and_or
tff(fact_6568_word__or__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),uminus_uminus(word(A),one_one(word(A)))) = uminus_uminus(word(A),one_one(word(A))) ) ).

% word_or_max
tff(fact_6569_word__bitwise__m1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),uminus_uminus(word(A),one_one(word(A)))),Xa) = uminus_uminus(word(A),one_one(word(A))) ) ).

% word_bitwise_m1_simps(4)
tff(fact_6570_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa)) ).

% or_nat_numerals(4)
tff(fact_6571_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_6572_word__or__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)) = uminus_uminus(word(A),one_one(word(A))) ) ).

% word_or_not
tff(fact_6573_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa)) ).

% or_nat_numerals(3)
tff(fact_6574_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_6575_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_6576_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_6577_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_6578_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)),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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(8)
tff(fact_6579_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)),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(9)
tff(fact_6580_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)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),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),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(10)
tff(fact_6581_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))),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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_bitwise_1_simps(7)
tff(fact_6582_word__not__dist_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Y)) ) ).

% word_not_dist(1)
tff(fact_6583_word__not__dist_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Y)) ) ).

% word_not_dist(2)
tff(fact_6584_word__log__esimps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),zero_zero(word(A))),Xa) = Xa ) ).

% word_log_esimps(9)
tff(fact_6585_word__log__esimps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),zero_zero(word(A))) = Xa ) ).

% word_log_esimps(3)
tff(fact_6586_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_6587_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_wz(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_6588_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_aj(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_6589_word__bw__assocs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: 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),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) ) ).

% word_bw_assocs(2)
tff(fact_6590_word__bw__comms_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Xa) ) ).

% word_bw_comms(2)
tff(fact_6591_word__bw__same_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Xa) = Xa ) ).

% word_bw_same(2)
tff(fact_6592_word__bw__lcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) ) ).

% word_bw_lcs(2)
tff(fact_6593_refines__option,axiom,
    ! [B: $tType,A: $tType,A2: option(A),A6: option(A),M1: heap_Time_Heap(B),M12: heap_Time_Heap(B),M22: fun(A,heap_Time_Heap(B)),M23: fun(A,heap_Time_Heap(B))] :
      ( ( A2 = A6 )
     => ( refine_Imp_refines(B,M1,M12)
       => ( ! [X: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),M22,X),aa(A,heap_Time_Heap(B),M23,X))
         => refine_Imp_refines(B,case_option(heap_Time_Heap(B),A,M1,M22,A2),case_option(heap_Time_Heap(B),A,M12,M23,A6)) ) ) ) ).

% refines_option
tff(fact_6594_TBOUND__option__case,axiom,
    ! [B: $tType,A: $tType,Ta: option(A),F: heap_Time_Heap(B),Bnd: nat,F3: fun(A,heap_Time_Heap(B)),Bnd2: fun(A,nat)] :
      ( ( ( Ta = none(A) )
       => time_TBOUND(B,F,Bnd) )
     => ( ! [X: A] :
            ( ( Ta = aa(A,option(A),some(A),X) )
           => time_TBOUND(B,aa(A,heap_Time_Heap(B),F3,X),aa(A,nat,Bnd2,X)) )
       => time_TBOUND(B,case_option(heap_Time_Heap(B),A,F,F3,Ta),case_option(nat,A,Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_6595_leao,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A),X6: word(A),Y7: word(A)] :
          ( ( W3 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X6),Y7) )
         => ( X6 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X6),W3) ) ) ) ).

% leao
tff(fact_6596_leoa,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Y: word(A)] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y) )
         => ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),Y) ) ) ) ).

% leoa
tff(fact_6597_word__ao__dist,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: 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_se1065995026697491101ons_or(word(A)),Xa),Y)),Z) = 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)),Xa),Z)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) ) ).

% word_ao_dist
tff(fact_6598_word__oa__dist,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: 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),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Z)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) ) ).

% word_oa_dist
tff(fact_6599_word__ao__dist2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) = 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)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Z)) ) ).

% word_ao_dist2
tff(fact_6600_word__ao__equiv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),W3: word(A)] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),W),W3) )
        <=> ( W3 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),W3) ) ) ) ).

% word_ao_equiv
tff(fact_6601_word__oa__dist2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Z)) ) ).

% word_oa_dist2
tff(fact_6602_uint__or,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_or
tff(fact_6603_word__wi__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),A2),B2)) ) ).

% word_wi_log_defs(3)
tff(fact_6604_le__word__or1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),C2)) ) ).

% le_word_or1
tff(fact_6605_le__word__or2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) ) ).

% le_word_or2
tff(fact_6606_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xa: option(A)] :
      ( case_option($o,A,(P),Q,Xa)
     => ( ( ( Xa = none(A) )
         => ~ (P) )
       => ~ ! [Y3: A] :
              ( ( Xa = aa(A,option(A),some(A),Y3) )
             => ~ aa(A,$o,Q,Y3) ) ) ) ).

% case_optionE
tff(fact_6607_word__plus__and__or__coroll,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y) = zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y) ) ) ) ).

% word_plus_and_or_coroll
tff(fact_6608_word__or__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_or_def
tff(fact_6609_word__xor__and__or,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y) = 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)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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),bit_ri4277139882892585799ns_not(word(A)),Xa)),Y)) ) ).

% word_xor_and_or
tff(fact_6610_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),Xa),Y)
        <=> case_option($o,A,$true,aTP_Lamp_xa(option(A),fun(A,$o),Y),Xa) ) ) ).

% less_eq_option_def
tff(fact_6611_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),Xa),Y)
        <=> case_option($o,A,$false,aTP_Lamp_xc(option(A),fun(A,$o),Xa),Y) ) ) ).

% less_option_def
tff(fact_6612_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),bit_se2239418461657761734s_mask(word(A),Nb)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_subsume
tff(fact_6613_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),Xa: B,Xs: list(B)] : map_filter(B,A,F,aa(list(B),list(B),cons(B,Xa),Xs)) = case_option(list(A),A,map_filter(B,A,F,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_xd(fun(B,option(A)),fun(list(B),fun(A,list(A))),F),Xs),aa(B,option(A),F,Xa)) ).

% map_filter_simps(1)
tff(fact_6614_case__option__rule,axiom,
    ! [A: $tType,B: $tType,Va2: option(A),P: assn,Fn: heap_Time_Heap(B),Q: fun(B,assn),Fs: fun(A,heap_Time_Heap(B))] :
      ( ( ( Va2 = none(A) )
       => hoare_hoare_triple(B,P,Fn,Q) )
     => ( ! [X: A] :
            ( ( Va2 = 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,Va2),Q) ) ) ).

% case_option_rule
tff(fact_6615_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,size_size(word(A)),Xa))
         => ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),Nb)
            <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb)
                | aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),Nb) ) )
            & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),Nb)
            <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),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)),Xa),Y)),Nb)
            <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb)
                <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),Nb) ) )
            & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),Nb)
            <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb) ) ) ) ) ).

% word_ops_nth_size
tff(fact_6616_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_6617_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_6618_or__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)
            | ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_6619_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,aa(nat,fun(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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_6620_word__ops__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),zero_zero(nat))
              | aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),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)),Xa),Y)),zero_zero(nat))
          <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),zero_zero(nat))
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),zero_zero(nat))
          <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),zero_zero(nat)) ) ) ) ).

% word_ops_lsb
tff(fact_6621_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)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),Nb))) = case_option(word(A),num,zero_zero(word(A)),aTP_Lamp_xe(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_6622_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_6623_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_6624_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_xf(num,option(num)),bit_take_bit_num(Nb,M)) ).

% take_bit_num_simps(3)
tff(fact_6625_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_6626_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_xf(num,option(num)),bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_6627_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_6628_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_6629_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)),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb))) = case_option(int,num,zero_zero(int),aTP_Lamp_xg(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_6630_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_6631_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,$o),Xa: option(A),R: fun(B,$o),F: B,G: fun(A,B)] :
      ( case_option($o,A,(P),Q,Xa)
     => ( ( (P)
         => aa(B,$o,R,F) )
       => ( ! [Q3: A] :
              ( aa(A,$o,Q,Q3)
             => aa(B,$o,R,aa(A,B,G,Q3)) )
         => aa(B,$o,R,case_option(B,A,F,G,Xa)) ) ) ) ).

% disjE_realizer2
tff(fact_6632_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)),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_6633_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(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_6634_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_xh(num,option(num)),bit_and_not_num(M,Nb)) ).

% and_not_num.simps(8)
tff(fact_6635_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_6636_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_6637_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_6638_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)),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_6639_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(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_6640_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_6641_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(bit0(Nb)) = aa(num,num,bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_6642_semiring__norm_I28_J,axiom,
    ! [Nb: num] : bitM(aa(num,num,bit1,Nb)) = aa(num,num,bit1,bit0(Nb)) ).

% semiring_norm(28)
tff(fact_6643_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_6644_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_xi(num,fun(nat,option(num)),M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_6645_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_xj(num,fun(nat,option(num)),M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_6646_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F: fun(nat,A),Va2: num] : case_nat(A,A2,F,aa(num,nat,numeral_numeral(nat),Va2)) = aa(nat,A,F,pred_numeral(Va2)) ).

% case_nat_numeral
tff(fact_6647_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F: fun(nat,A),Va2: num,Nb: nat] : case_nat(A,A2,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),Va2)),Nb)) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(Va2)),Nb)) ).

% case_nat_add_eq_if
tff(fact_6648_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X2: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ).

% old.nat.simps(5)
tff(fact_6649_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F1,F22,Nat)) = case_nat(A,aa(B,A,H,F1),aa(fun(nat,B),fun(nat,A),aTP_Lamp_xk(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_6650_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A)] : case_nat(A,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_6651_nth__Cons,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Nb) = case_nat(A,Xa,nth(A,Xs),Nb) ).

% nth_Cons
tff(fact_6652_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xa: A,F: fun(nat,A),Nb: nat] :
      case_nat(A,Xa,F,Nb) = $ite(Nb = zero_zero(nat),Xa,aa(nat,A,F,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_6653_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_xl(nat,option(num)),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_6654_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(Nb,M) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_xp(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),Nb),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_6655_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_xq(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_inverse_code
tff(fact_6656_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_6657_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_xr(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_6658_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_xs(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_6659_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(num,B),F32: fun(num,B),Num: num] : aa(B,A,H,case_num(B,F1,F22,F32,Num)) = case_num(A,aa(B,A,H,F1),aa(fun(num,B),fun(num,A),aTP_Lamp_xt(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22),aa(fun(num,B),fun(num,A),aTP_Lamp_xt(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32),Num) ).

% num.case_distrib
tff(fact_6660_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_6661_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q4: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q4) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q4) ) ).

% quotient_of_denom_pos
tff(fact_6662_max__Suc1,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),M) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_xu(nat,fun(nat,nat),Nb),M) ).

% max_Suc1
tff(fact_6663_max__Suc2,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,Nb)) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_xv(nat,fun(nat,nat),Nb),M) ).

% max_Suc2
tff(fact_6664_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),I: nat,Va2: A] : list_update(A,aa(list(A),list(A),cons(A,Xa),Xs),I,Va2) = case_nat(list(A),aa(list(A),list(A),cons(A,Va2),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_xw(A,fun(list(A),fun(A,fun(nat,list(A)))),Xa),Xs),Va2),I) ).

% list_update.simps(2)
tff(fact_6665_drop__Cons,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] : drop(A,Nb,aa(list(A),list(A),cons(A,Xa),Xs)) = case_nat(list(A),aa(list(A),list(A),cons(A,Xa),Xs),aTP_Lamp_xx(list(A),fun(nat,list(A)),Xs),Nb) ).

% drop_Cons
tff(fact_6666_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_kx(nat,nat),aa(nat,nat,minus_minus(nat,M),Nb)) ).

% diff_Suc
tff(fact_6667_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_6668_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_6669_take__Cons,axiom,
    ! [A: $tType,Nb: nat,Xa: A,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),cons(A,Xa),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_xy(A,fun(list(A),fun(nat,list(A))),Xa),Xs),Nb) ).

% take_Cons
tff(fact_6670_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(uminus_uminus(rat,P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_xz(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_uminus_code
tff(fact_6671_rat__less__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P3),Q4)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_yb(rat,fun(int,fun(int,$o)),Q4)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_6672_rat__floor__code,axiom,
    ! [P3: rat] : archim6421214686448440834_floor(rat,P3) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),divide_divide(int)),quotient_of(P3)) ).

% rat_floor_code
tff(fact_6673_rat__abs__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_yc(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_abs_code
tff(fact_6674_rat__less__eq__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P3),Q4)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_ye(rat,fun(int,fun(int,$o)),Q4)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_6675_rat__plus__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_yg(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).

% rat_plus_code
tff(fact_6676_rat__minus__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,minus_minus(rat,P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_yi(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).

% rat_minus_code
tff(fact_6677_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_6678_normalize__negative,axiom,
    ! [Q4: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q4),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q4)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),uminus_uminus(int,P3)),uminus_uminus(int,Q4))) ) ) ).

% normalize_negative
tff(fact_6679_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q4: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q4) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q4) ) ).

% normalize_denom_pos
tff(fact_6680_normalize__crossproduct,axiom,
    ! [Q4: int,S2: int,P3: int,R2: int] :
      ( ( Q4 != zero_zero(int) )
     => ( ( S2 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q4)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S2)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_6681_rat__times__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_yk(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).

% rat_times_code
tff(fact_6682_rat__divide__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ym(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).

% rat_divide_code
tff(fact_6683_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_yn(A,fun(list(A),A))),aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_dw(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_6684_sofl__test,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,Xa)),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)),Xa),Y)) )
        <=> ( bit_se4197421643247451524op_bit(word(A),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),Xa)),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)),Xa),Y)),Xa)),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)),Xa),Y)),Y))) = zero_zero(word(A)) ) ) ) ).

% sofl_test
tff(fact_6685_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_6686_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_6687_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_6688_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_6689_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_6690_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_6691_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_6692_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,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_6693_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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_6694_div__half__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),modulo_modulo(word(A),Xa,Y)) = $let(
                q: word(A),
                q:= bit_se4730199178511100633sh_bit(word(A),one_one(nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),bit_se4197421643247451524op_bit(word(A),one_one(nat),Xa)),Y)),
                $let(
                  r: word(A),
                  r:= aa(word(A),word(A),minus_minus(word(A),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                  $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),r),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),q),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),r),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),q),r)) ) ) ) ) ) ).

% div_half_word
tff(fact_6695_take__bit__num__def,axiom,
    ! [Nb: nat,M: num] :
      bit_take_bit_num(Nb,M) = $ite(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat),none(num),aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(num,nat,numeral_numeral(nat),M))))) ).

% take_bit_num_def
tff(fact_6696_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_6697_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_6698_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se4197421643247451524op_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_6699_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se4197421643247451524op_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_6700_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_6701_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_6702_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb),uminus_uminus(int,aa(num,int,numeral_numeral(int),bit0(K)))) = bit_se4197421643247451524op_bit(int,Nb,uminus_uminus(int,aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_6703_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),uminus_uminus(A,aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,Nb,uminus_uminus(A,aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_6704_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb),uminus_uminus(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = bit_se4197421643247451524op_bit(int,Nb,uminus_uminus(int,aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_6705_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_6706_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_6707_unat__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se4197421643247451524op_bit(word(A),Nb,W)) = bit_se4197421643247451524op_bit(nat,Nb,aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_drop_bit_eq
tff(fact_6708_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_6709_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_6710_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6711_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_6712_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_6713_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_6714_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),one_one(nat))
     => ( num_of_nat(Nb) = one2 ) ) ).

% num_of_nat_One
tff(fact_6715_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_6716_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_6717_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_6718_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$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_6719_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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_yo(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_6720_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      num_of_nat(aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb),inc(num_of_nat(Nb)),one2) ).

% num_of_nat.simps(2)
tff(fact_6721_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),uminus_uminus(A,one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_6722_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F,aa(nat,nat,suc,Nb),W) = code_T2661198915054445665ts_aux(A,F,Nb,
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),W)),
              $ite(aa(nat,$o,F,Nb),one_one(word(A)),zero_zero(word(A))))) ) ).

% set_bits_aux_Suc
tff(fact_6723_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F,Nb,W) = $ite(
            Nb = zero_zero(nat),
            W,
            $let(
              n: nat,
              n:= aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),
              code_T2661198915054445665ts_aux(A,F,n,
                aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),W)),
                  $ite(aa(nat,$o,F,n),one_one(word(A)),zero_zero(word(A))))) ) ) ) ).

% set_bits_aux_rec
tff(fact_6724_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se4730199178511100633sh_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_6725_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se4730199178511100633sh_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_6726_set__bits__aux__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),W: word(A)] : code_T2661198915054445665ts_aux(A,F,zero_zero(nat),W) = W ) ).

% set_bits_aux_0
tff(fact_6727_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_6728_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_6729_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,aa(nat,fun(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_6730_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,M,Q4)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q4),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_6731_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_6732_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
        <=> ( ! [N6: nat] :
                ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(B),B2),N6)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N6),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,bit_se5641148757651400278ts_bit(word(C),C2),N6) ) )
            & ! [M5: nat] :
                ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),A2),M5)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),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),M5),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_6733_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ! [N2: nat,M3: nat] :
              ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(B),B2),N2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,bit_se5641148757651400278ts_bit(word(C),C2),N2) ) )
              & ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),A2),M3)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$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_6734_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ( ! [N2: nat] :
                ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(B),B2),N2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,bit_se5641148757651400278ts_bit(word(C),C2),N2) ) )
            & ! [M3: nat] :
                ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),A2),M3)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$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_6735_case__prod__app,axiom,
    ! [D6: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D6,A))),Xa: product_prod(B,C),Y: D6] : aa(D6,A,aa(product_prod(B,C),fun(D6,A),aa(fun(B,fun(C,fun(D6,A))),fun(product_prod(B,C),fun(D6,A)),product_case_prod(B,C,fun(D6,A)),F),Xa),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D6,fun(B,fun(C,A)),aTP_Lamp_uf(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),F),Y)),Xa) ).

% case_prod_app
tff(fact_6736_subset__eq__mset__impl_Oelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: option($o)] :
      ( ( subset_eq_mset_impl(A,Xa,Xaa) = Y )
     => ( ( ( Xa = nil(A) )
         => ( Y != aa($o,option($o),some($o),Xaa != nil(A)) ) )
       => ~ ! [X: A,Xs2: list(A)] :
              ( ( Xa = aa(list(A),list(A),cons(A,X),Xs2) )
             => ( Y != case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),aa(fun(list(A),fun(product_prod(A,list(A)),option($o))),fun(product_prod(list(A),product_prod(A,list(A))),option($o)),product_case_prod(list(A),product_prod(A,list(A)),option($o)),aTP_Lamp_yq(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs2)),extract(A,aa(A,fun(A,$o),fequal(A),X),Xaa)) ) ) ) ) ).

% subset_eq_mset_impl.elims
tff(fact_6737_subset__eq__mset__impl_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : subset_eq_mset_impl(A,nil(A),Ys) = aa($o,option($o),some($o),Ys != nil(A)) ).

% subset_eq_mset_impl.simps(1)
tff(fact_6738_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),R: $o,Xa: A,Y: B] :
      ( ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
        & (R) )
     => ( (R)
        & ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Y)
         => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Y) ) ) ) ).

% predicate2D_conj
tff(fact_6739_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F: fun(fun(A,B),C),G: C] :
          ( ! [X: fun(A,B)] : aa(fun(A,B),C,F,X) = G
         => ( aa(fun(A,B),C,F,aTP_Lamp_yr(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_6740_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),aTP_Lamp_ys(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P)) ).

% eq_subset
tff(fact_6741_subset__eq__mset__impl_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ys: list(A)] : subset_eq_mset_impl(A,aa(list(A),list(A),cons(A,Xa),Xs),Ys) = case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),aa(fun(list(A),fun(product_prod(A,list(A)),option($o))),fun(product_prod(list(A),product_prod(A,list(A))),option($o)),product_case_prod(list(A),product_prod(A,list(A)),option($o)),aTP_Lamp_yq(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs)),extract(A,aa(A,fun(A,$o),fequal(A),Xa),Ys)) ).

% subset_eq_mset_impl.simps(2)
tff(fact_6742_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P3) = P3 ).

% case_prod_Pair_iden
tff(fact_6743_subset__eq__mset__impl_Opelims,axiom,
    ! [A: $tType,Xa: list(A),Xaa: list(A),Y: option($o)] :
      ( ( subset_eq_mset_impl(A,Xa,Xaa) = Y )
     => ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xaa))
       => ( ( ( Xa = nil(A) )
           => ( ( Y = aa($o,option($o),some($o),Xaa != nil(A)) )
             => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xaa)) ) )
         => ~ ! [X: A,Xs2: list(A)] :
                ( ( Xa = aa(list(A),list(A),cons(A,X),Xs2) )
               => ( ( Y = case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),aa(fun(list(A),fun(product_prod(A,list(A)),option($o))),fun(product_prod(list(A),product_prod(A,list(A))),option($o)),product_case_prod(list(A),product_prod(A,list(A)),option($o)),aTP_Lamp_yq(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs2)),extract(A,aa(A,fun(A,$o),fequal(A),X),Xaa)) )
                 => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs2)),Xaa)) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
tff(fact_6744_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,Nb),K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_concat_bit(Nb,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))),L))) ).

% concat_bit_Suc
tff(fact_6745_concat__bit__0,axiom,
    ! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).

% concat_bit_0
tff(fact_6746_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: 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_6747_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_concat_bit(Nb,K,L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_6748_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_concat_bit(Nb,K,L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_6749_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : bit_concat_bit(Nb,zero_zero(int),L) = bit_se4730199178511100633sh_bit(int,Nb,L) ).

% concat_bit_of_zero_1
tff(fact_6750_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,bit_concat_bit(M,K,L)),Nb)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6751_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [Ta: word(A),U: word(B),Va2: word(C)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Ta)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),Va2)))
         => ( ( aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),Va2) = word_split(A,B,C,Ta) )
           => ( Ta = word_cat(B,C,A,U,Va2) ) ) ) ) ).

% word_cat_split_size
tff(fact_6752_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(A),U: word(B),Va2: word(C)] :
          ( ( W = word_cat(B,C,A,U,Va2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),Va2))),aa(word(A),nat,size_size(word(A)),W))
           => ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),Va2) ) ) ) ) ).

% word_split_cat_alt
tff(fact_6753_word__cat__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A2: word(B),B2: word(A)] : word_cat(B,A,A,A2,B2) = B2 ) ).

% word_cat_id
tff(fact_6754_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(word(A),nat,size_size(word(A)),word_cat(B,C,A,A2,B2)))
            & $ite(aa(nat,$o,aa(nat,fun(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_6755_word__cat__split__alt,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B)
        & type_len(C) )
     => ! [W: word(A),U: word(B),Va2: word(C)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),Va2)))
         => ( ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),Va2) )
           => ( word_cat(B,C,A,U,Va2) = W ) ) ) ) ).

% word_cat_split_alt
tff(fact_6756_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A2: word(A),Nb: nat,C2: word(B),B2: word(C)] :
          ( ( A2 = aa(word(B),word(A),slice2(B,A,Nb),C2) )
         => ( ( B2 = aa(word(B),word(C),slice2(B,C,zero_zero(nat)),C2) )
           => ( ( Nb = aa(word(C),nat,size_size(word(C)),B2) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),C2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(C),nat,size_size(word(C)),B2)))
               => ( word_cat(A,C,B,A2,B2) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_6757_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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_6758_int__sdiv__simps_I2_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,zero_zero(int)) = zero_zero(int) ).

% int_sdiv_simps(2)
tff(fact_6759_sdiv__int__0__div,axiom,
    ! [Xa: int] : signed7115095781618012415divide(int,zero_zero(int),Xa) = zero_zero(int) ).

% sdiv_int_0_div
tff(fact_6760_sdiv__int__div__0,axiom,
    ! [Xa: int] : signed7115095781618012415divide(int,Xa,zero_zero(int)) = zero_zero(int) ).

% sdiv_int_div_0
tff(fact_6761_slice__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat] : aa(word(B),word(A),slice2(B,A,Nb),zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice_0
tff(fact_6762_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_6763_int__sdiv__negated__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A2,B2) = uminus_uminus(int,A2) )
      <=> ( B2 = uminus_uminus(int,one_one(int)) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_6764_slice__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ta: word(A)] : aa(word(A),word(A),slice2(A,A,zero_zero(nat)),Ta) = Ta ) ).

% slice_id
tff(fact_6765_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) = aa(word(B),word(A),slice2(B,A,zero_zero(nat)),W) ) ).

% ucast_slice
tff(fact_6766_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_6767_slice__cat2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A2: word(B),Ta: word(A)] : aa(word(A),word(A),slice2(A,A,zero_zero(nat)),word_cat(B,A,A,A2,Ta)) = Ta ) ).

% slice_cat2
tff(fact_6768_slice__cat1,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [A2: word(A),B2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(B),nat,size_size(word(B)),B2))),aa(word(C),nat,size_size(word(C)),word_cat(A,B,C,A2,B2)))
         => ( aa(word(C),word(A),slice2(C,A,aa(word(B),nat,size_size(word(B)),B2)),word_cat(A,B,C,A2,B2)) = A2 ) ) ) ).

% slice_cat1
tff(fact_6769_split__slices,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(C),U: word(A),Va2: word(B)] :
          ( ( word_split(C,A,B,W) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),U),Va2) )
         => ( ( U = aa(word(C),word(A),slice2(C,A,aa(word(B),nat,size_size(word(B)),Va2)),W) )
            & ( Va2 = aa(word(C),word(B),slice2(C,B,zero_zero(nat)),W) ) ) ) ) ).

% split_slices
tff(fact_6770_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed7115095781618012415divide(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),A2)),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sdiv_word_max
tff(fact_6771_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M11: nat] :
        ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M5)
         => ! [N6: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N6)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,X5,M5)),aa(nat,real,X5,N6)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_6772_hash__code__option__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),Xa: A] : hash_h1887023736457453652option(A,H_a,aa(A,option(A),some(A),Xa)) = 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,Xa)),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_6773_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M10: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M3)
               => ! [N2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M3)),aa(nat,A,X5,N2)))),E2) ) ) ) ) ) ).

% CauchyD
tff(fact_6774_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M13: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M13),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M13),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M2)),aa(nat,A,X5,N)))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_6775_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M11: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M5)
                 => ! [N6: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N6)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M5)),aa(nat,A,X5,N6)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_6776_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : modulo_modulo(word(A),uminus_uminus(word(A),one_one(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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_minus_1_minus_numeral
tff(fact_6777_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),uminus_uminus(word(A),one_one(word(A)))),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_minus_1_minus_numeral
tff(fact_6778_take__bit__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),type_len0_len_of(A,type2(A))),W) = W ) ).

% take_bit_length_eq
tff(fact_6779_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( bit_se4197421643247451524op_bit(word(A),Nb,W) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_6780_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( bit_se4730199178511100633sh_bit(word(A),Nb,W) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_6781_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_6782_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),uminus_uminus(A,one_one(A)),one_one(A)) ) ) ).

% signed_1
tff(fact_6783_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_6784_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_6785_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(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_6786_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_6787_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,$o,aa(nat,fun(nat,$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_6788_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_6789_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_6790_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),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_6791_uint__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(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_6792_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),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_6793_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb) = zero_zero(word(A)) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb) ) ) ).

% exp_eq_zero_iff
tff(fact_6794_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_6795_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_6796_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_6797_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_6798_uint__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,semiring_1_unsigned(A,int),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),Bin))) ) ).

% uint_bintrunc_neg
tff(fact_6799_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_numeral_numeral
tff(fact_6800_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_6801_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_6802_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_6803_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_6804_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),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_6805_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_6806_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_6807_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),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_6808_unat__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),Bin)))) ) ).

% unat_bintrunc_neg
tff(fact_6809_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),W))),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,uminus_uminus(int,aa(num,int,numeral_numeral(int),W))),Nb) ) ) ) ).

% bit_neg_numeral_word_iff
tff(fact_6810_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_6811_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_6812_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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_6813_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),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb))))) ) ).

% unsigned_neg_numeral
tff(fact_6814_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)),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_Suc_minus_numeral
tff(fact_6815_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)),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_minus_numeral
tff(fact_6816_sint__sbintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : ring_1_signed(A,int,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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),Bin))) ) ).

% sint_sbintrunc_neg
tff(fact_6817_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_6818_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_6819_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),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_6820_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_numeral_minus_numeral
tff(fact_6821_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_minus_numeral_minus_numeral
tff(fact_6822_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : modulo_modulo(word(A),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_minus_numeral_minus_numeral
tff(fact_6823_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),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_numeral_minus_numeral
tff(fact_6824_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : modulo_modulo(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))),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_6825_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(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_6826_signed__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Nb: num] : ring_1_signed(B,A,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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),Nb)))) ) ).

% signed_neg_numeral
tff(fact_6827_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),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_Suc_minus_numeral
tff(fact_6828_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),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_minus_numeral
tff(fact_6829_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),uminus_uminus(word(A),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(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_6830_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),uminus_uminus(word(A),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),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_6831_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),uminus_uminus(word(A),one_one(word(A)))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_minus_1_numeral
tff(fact_6832_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : modulo_modulo(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_6833_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),aa(word(B),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,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_slice
tff(fact_6834_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( aa(int,word(C),ring_1_of_int(word(C)),bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),B2),aa(word(A),int,semiring_1_unsigned(A,int),A2))) = aa(int,word(C),ring_1_of_int(word(C)),bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),D),aa(word(A),int,semiring_1_unsigned(A,int),C2))) )
          <=> ( ( B2 = D )
              & ( A2 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_6835_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_6836_uint__word__arith__bintrs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(1)
tff(fact_6837_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Y: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ) ) ).

% ucast_ucast_add
tff(fact_6838_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_6839_num__abs__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(num,word(A),numeral_numeral(word(A)),Xa) = 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),Xa))) ) ).

% num_abs_bintr
tff(fact_6840_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(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_6841_uint__word__arith__bintrs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(3)
tff(fact_6842_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( word_cat(A,B,C,A2,B2) = word_cat(A,B,C,C2,D) )
          <=> ( ( A2 = C2 )
              & ( B2 = D ) ) ) ) ) ).

% word_cat_inj
tff(fact_6843_word__cat__bin_H,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [Va2: word(B),W: word(C)] : word_cat(B,C,A,Va2,W) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),bit_se4730199178511100633sh_bit(word(A),type_len0_len_of(C,type2(C)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Va2))),aa(word(C),word(A),semiring_1_unsigned(C,word(A)),W)) ) ).

% word_cat_bin'
tff(fact_6844_word__cat__eq_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A2: word(B),B2: word(C)] : word_cat(B,C,A,A2,B2) = aa(int,word(A),ring_1_of_int(word(A)),bit_concat_bit(type_len0_len_of(C,type2(C)),aa(word(C),int,semiring_1_unsigned(C,int),B2),aa(word(B),int,semiring_1_unsigned(B,int),A2))) ) ).

% word_cat_eq'
tff(fact_6845_bit__word__cat__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [Va2: word(B),W: word(C),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),word_cat(B,C,A,Va2,W)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,aa(nat,fun(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),Va2),aa(nat,nat,minus_minus(nat,Nb),type_len0_len_of(C,type2(C))))) ) ) ) ).

% bit_word_cat_iff
tff(fact_6846_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),bit_se4197421643247451524op_bit(word(A),Nb,W)) = bit_se4197421643247451524op_bit(word(B),Nb,aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W)) ) ) ) ).

% ucast_drop_bit_eq
tff(fact_6847_unsigned__drop__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_un5681908812861735899ations(A) )
     => ! [Nb: nat,W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),bit_se4197421643247451524op_bit(word(B),Nb,W)) = bit_se4197421643247451524op_bit(A,Nb,aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),aa(word(B),A,semiring_1_unsigned(B,A),W))) ) ).

% unsigned_drop_bit_eq
tff(fact_6848_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat,W: word(A),M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),code_T2661198915054445665ts_aux(A,F,Nb,W)),M)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb),aa(nat,$o,F,M),aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,M),Nb))) ) ) ) ).

% bit_set_bit_aux
tff(fact_6849_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_6850_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_6851_unsigned__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [K: int] : aa(word(B),A,semiring_1_unsigned(B,A),aa(int,word(B),ring_1_of_int(word(B)),K)) = aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),K))) ) ).

% unsigned_of_int
tff(fact_6852_bintr__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ) ) ).

% bintr_uint
tff(fact_6853_unsigned__ucast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_se359711467146920520ations(A)
        & type_len(B) )
     => ! [W: word(C)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W)) = aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),aa(word(C),A,semiring_1_unsigned(C,A),W)) ) ).

% unsigned_ucast_eq
tff(fact_6854_unsigned__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Nb: nat] : aa(word(B),A,semiring_1_unsigned(B,A),aa(nat,word(B),semiring_1_of_nat(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))),Nb)) ) ).

% unsigned_of_nat
tff(fact_6855_uint__word__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] : aa(word(A),int,semiring_1_unsigned(A,int),aa(int,word(A),ring_1_of_int(word(A)),K)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K) ) ).

% uint_word_of_int_eq
tff(fact_6856_uint__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),W) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),ring_1_signed(A,int,W)) ) ).

% uint_sint
tff(fact_6857_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,aa(nat,fun(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_6858_word__of__int__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),K) = aa(int,word(A),ring_1_of_int(word(A)),L) )
        <=> ( 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_eq_iff
tff(fact_6859_word__of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),M) = aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) )
        <=> ( 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_eq_iff
tff(fact_6860_word__of__int__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(int,fun(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_6861_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(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_6862_wi__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: int] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),W)) = aa(int,word(A),ring_1_of_int(word(A)),W) ) ) ) ).

% wi_bintr
tff(fact_6863_word__of__int__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_eq_iff
tff(fact_6864_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Nb),W) = W ) ) ) ).

% take_bit_word_beyond_length_eq
tff(fact_6865_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),Nb)) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_6866_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( ring_1_signed(A,word(B),Xa) = ring_1_signed(A,word(B),Y) )
          <=> ( Xa = Y ) ) ) ) ).

% up_scast_inj_eq
tff(fact_6867_ucast__ucast__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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)),Xa)) = Xa ) ) ) ).

% ucast_ucast_id
tff(fact_6868_ucast__less__ucast__weak,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% ucast_less_ucast_weak
tff(fact_6869_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),nat,semiring_1_unsigned(B,nat),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% unat_ucast_up_simp
tff(fact_6870_eq__ucast__ucast__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( Xa = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
           => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = Y ) ) ) ) ).

% eq_ucast_ucast_eq
tff(fact_6871_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono_le
tff(fact_6872_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
          <=> ( Xa = Y ) ) ) ) ).

% up_ucast_inj_eq
tff(fact_6873_ucast__ucast__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(C)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(C,type2(C))),type_len0_len_of(A,type2(A)))
             => ( Xa = aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y) ) ) ) ) ) ).

% ucast_ucast_eq
tff(fact_6874_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y) ) ) ) ).

% ucast_le_ucast
tff(fact_6875_up__ucast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(B)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( Xa = Y ) ) ) ) ).

% up_ucast_inj
tff(fact_6876_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono
tff(fact_6877_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% ucast_less_ucast
tff(fact_6878_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),Xa),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa)),Y) ) ) ) ).

% less_ucast_ucast_less
tff(fact_6879_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat,Xa: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Xa),bit_se2239418461657761734s_mask(word(B),Nb))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) ) ) ) ).

% ucast_mask_drop
tff(fact_6880_degenerate__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
         => ( ( Xa = zero_zero(word(A)) )
            | ( Xa = one_one(word(A)) ) ) ) ) ).

% degenerate_word
tff(fact_6881_size__0__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Va2: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = zero_zero(nat) )
         => ( W = Va2 ) ) ) ).

% size_0_same
tff(fact_6882_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),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_6883_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( bit_se2239418461657761734s_mask(word(A),Nb) = uminus_uminus(word(A),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_6884_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_6885_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_6886_signed__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B)] : ring_1_signed(B,A,aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),W)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,type_len0_len_of(B,type2(B))),aa(A,A,bit_ri4277139882892585799ns_not(A),ring_1_signed(B,A,W))) ) ).

% signed_not_eq
tff(fact_6887_unsigned__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),W)) = aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(word(B),A,semiring_1_unsigned(B,A),W))) ) ).

% unsigned_not_eq
tff(fact_6888_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_6889_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_6890_uint__word__arith__bintrs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),uminus_uminus(word(A),A2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),uminus_uminus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2))) ) ).

% uint_word_arith_bintrs(4)
tff(fact_6891_size__word_Orep__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),nat,size_size(word(A)),Xa) = type_len0_len_of(A,type2(A)) ) ).

% size_word.rep_eq
tff(fact_6892_word__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,size_size(word(A)),W) = type_len0_len_of(A,type2(A)) ) ).

% word_size
tff(fact_6893_up__scast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(B)] :
          ( ( ring_1_signed(B,word(A),Xa) = ring_1_signed(B,word(A),Y) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),Xa)),type_len0_len_of(A,type2(A)))
           => ( Xa = Y ) ) ) ) ).

% up_scast_inj
tff(fact_6894_neg__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),Nb)
        <=> ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_test_bit
tff(fact_6895_max__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),uminus_uminus(word(A),one_one(word(A)))),Nb)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ).

% max_test_bit
tff(fact_6896_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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),type_len0_len_of(A,type2(A)))
            & ( Nb = zero_zero(nat) ) ) ) ) ).

% test_bit_1'
tff(fact_6897_not__bit__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),type_len0_len_of(A,type2(A))) ) ).

% not_bit_length
tff(fact_6898_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,$o,aa(nat,fun(nat,$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_6899_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,$o,aa(nat,fun(nat,$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_6900_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ).

% bin_nth_uint_imp
tff(fact_6901_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(B,type2(B)))
            & aa(nat,$o,aa(nat,fun(nat,$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_6902_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,$o,aa(nat,fun(nat,$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_6903_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,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_ucast
tff(fact_6904_test__bit__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(int,word(A),ring_1_of_int(word(A)),Xa)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,Xa),Nb) ) ) ) ).

% test_bit_wi
tff(fact_6905_word__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
        <=> ! [N6: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N6),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),N6)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Y),N6) ) ) ) ) ).

% word_eq_iff
tff(fact_6906_test__bit__conj__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),M: nat] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) )
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),M) ) ) ).

% test_bit_conj_lt
tff(fact_6907_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,$o,aa(nat,fun(nat,$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_6908_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ).

% bit_imp_le_length
tff(fact_6909_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(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_6910_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
           => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),minus_minus(word(A),Xa),Y)) = aa(word(B),word(B),minus_minus(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_sub_ucast
tff(fact_6911_uint__word__arith__bintrs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(2)
tff(fact_6912_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_6913_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(nat,$o,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_6914_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_6915_unsigned__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [Nb: nat,W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),bit_se4730199178511100633sh_bit(word(B),Nb,W)) = aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),bit_se4730199178511100633sh_bit(A,Nb,aa(word(B),A,semiring_1_unsigned(B,A),W))) ) ).

% unsigned_push_bit_eq
tff(fact_6916_word__split__bin_H,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(C)] : word_split(C,A,B,W) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),aa(word(C),word(A),semiring_1_unsigned(C,word(A)),bit_se4197421643247451524op_bit(word(C),type_len0_len_of(B,type2(B)),W))),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W)) ) ).

% word_split_bin'
tff(fact_6917_unsigned__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,aa(A,fun(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_6918_not__degenerate__imp__2__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,aa(nat,fun(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_6919_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(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)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% of_nat_inj
tff(fact_6920_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(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)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
             => ( Xa = Y ) ) ) ) ) ).

% word_of_nat_inj
tff(fact_6921_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ~ ! [N: nat] :
              ( ( Xa = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
             => ~ aa(nat,$o,aa(nat,fun(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_6922_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W4: word(A)] :
        ? [N: nat] :
          ( ( W4 = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
          & aa(nat,$o,aa(nat,fun(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_6923_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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_6924_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ) ) ) ).

% word_power_increasing
tff(fact_6925_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb) = zero_zero(word(A)) ) ) ) ).

% power_overflow
tff(fact_6926_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,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_w2p
tff(fact_6927_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ).

% nth_w2p_same
tff(fact_6928_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_6929_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_6930_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_6931_num__abs__sbintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(num,word(A),numeral_numeral(word(A)),Xa) = 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),Xa))) ) ).

% num_abs_sbintr
tff(fact_6932_horner__sum__uint__exp__Cons__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Ws: list(word(A))] : aa(list(word(A)),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)))),aa(list(word(A)),list(word(A)),cons(word(A),W),Ws)) = bit_concat_bit(type_len0_len_of(A,type2(A)),aa(word(A),int,semiring_1_unsigned(A,int),W),aa(list(word(A)),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)) ) ).

% horner_sum_uint_exp_Cons_eq
tff(fact_6933_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_6934_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_6935_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_6936_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,aa(nat,fun(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_6937_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,$o,aa(nat,fun(nat,$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_6938_sint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,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))),uminus_uminus(int,ring_1_signed(A,int,A2))) ) ).

% sint_word_ariths(4)
tff(fact_6939_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_6940_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_6941_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_6942_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_6943_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)))
         => aa(nat,$o,aa(nat,fun(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_6944_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
       => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = aa(num,int,numeral_numeral(int),bit0(one2)) ) ) ) ).

% uint_2_id
tff(fact_6945_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => aa(int,$o,aa(int,fun(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_6946_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ) ) ) ).

% power_le_mono
tff(fact_6947_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% two_power_increasing
tff(fact_6948_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_6949_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
           => aa(word(A),$o,aa(word(A),fun(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)),Xa)) ) ) ) ).

% of_nat_mono_maybe
tff(fact_6950_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
            <=> aa(word(A),$o,aa(word(A),fun(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)),Xa)) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_6951_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xa: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ! [N6: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N6) = Xa )
                & aa(nat,$o,aa(nat,fun(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_6952_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,aa(nat,fun(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_6953_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xa: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ~ ? [N6: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N6) = Xa )
                & aa(nat,$o,aa(nat,fun(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_6954_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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)),Xa)) = Xa ) ) ) ).

% unat_of_nat_len
tff(fact_6955_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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),Xa) = Nb )
          <=> ( Xa = aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_6956_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) != one_one(nat) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)))
           => ( ( Xa = zero_zero(word(A)) )
              | ( Xa = one_one(word(A)) ) ) ) ) ) ).

% x_less_2_0_1'
tff(fact_6957_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,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% test_bit_2p
tff(fact_6958_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)) ) ) ).

% word_1_le_power
tff(fact_6959_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)) )
        <=> ? [Q6: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q6),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_6960_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),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_6961_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_6962_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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)),Xa)) = aa(nat,word(B),semiring_1_of_nat(word(B)),Xa) ) ) ) ).

% ucast_of_nat_small
tff(fact_6963_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ).

% p2_gt_0
tff(fact_6964_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_6965_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)),uminus_uminus(int,I)) ) ).

% word_of_int_minus
tff(fact_6966_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,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,W)),zero_zero(int)) ) ) ).

% bit_last_iff
tff(fact_6967_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_6968_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_6969_unat__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = modulo_modulo(nat,Xa,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_6970_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_6971_ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),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_6972_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_6973_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_6974_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_6975_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N3: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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_6976_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,$o,aa(nat,fun(nat,$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,aa(nat,fun(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_6977_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono_le
tff(fact_6978_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_6979_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Nb),W) = W )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
            | aa(word(A),$o,aa(word(A),fun(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_6980_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_6981_msb0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),I: nat] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              | aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),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)),Xa),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              <=> aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) ) ) ).

% msb0
tff(fact_6982_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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),Xa)),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)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem
tff(fact_6983_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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),Xa)),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)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem'
tff(fact_6984_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,aa(nat,fun(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_6985_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,$o,aa(nat,fun(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_6986_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_6987_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_6988_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),one_one(word(A))))
          <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),one_one(word(A))) = one_one(word(A)) ) ) ) ) ).

% bool_mask'
tff(fact_6989_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),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_6990_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_6991_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,Xa)),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_6992_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ~ ! [N: int] :
              ( ( Xa = aa(int,word(A),ring_1_of_int(word(A)),N) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
               => ~ aa(int,$o,aa(int,fun(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_6993_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Y: int] :
          ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
              & aa(int,$o,aa(int,fun(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)),Xa) = aa(int,word(A),ring_1_of_int(word(A)),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% word_of_int_inj
tff(fact_6994_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_mono_le'
tff(fact_6995_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 )
        <=> ? [Q6: 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),Q6),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_6996_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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),Xa)),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)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_mult_lem
tff(fact_6997_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [B2: word(A),F: word(B)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),F)),B2)
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(B),nat,semiring_1_unsigned(B,nat),F)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_6998_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),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_6999_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa)),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_7000_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,F: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F)),Nb)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),F),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% unat_ucast_less_no_overflow
tff(fact_7001_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,F: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F)),Nb)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),F),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% unat_ucast_less_no_overflow_simp
tff(fact_7002_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(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_7003_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat,M: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),Nb)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M) ) ) ) ) ).

% nth_bounded
tff(fact_7004_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,P3: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => ( ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N11)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),P3),N11) ) )
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$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_7005_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
            & ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N11)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),P3),N11) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_7006_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(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),Xa)),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)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_add_lem
tff(fact_7007_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_7008_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_7009_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Nb)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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_7010_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_7011_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(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_7012_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(int,fun(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_7013_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ).

% power_2_ge_iff
tff(fact_7014_word__power__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(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_7015_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),uminus_uminus(word(A),A2)) = modulo_modulo(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_7016_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_7017_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))
          <=> aa(word(A),$o,aa(word(A),fun(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_7018_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(6)
tff(fact_7019_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,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_7020_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_7021_sint__1__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
           => ( ( A2 = zero_zero(word(A)) )
             => ( ring_1_signed(A,int,A2) != zero_zero(int) ) ) )
         => ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
             => ( ( A2 = one_one(word(A)) )
               => ( ring_1_signed(A,int,one_one(word(A))) != uminus_uminus(int,one_one(int)) ) ) )
           => ~ ( aa(nat,$o,aa(nat,fun(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_7022_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_7023_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(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),Xa)),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)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_mult_lem
tff(fact_7024_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_7025_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_7026_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),modulo_modulo(word(A),Xa,aa(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),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),aa(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_7027_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_7028_msb1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              | aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),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)),Xa),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              <=> aa(nat,$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,$o,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),Xa),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% msb1
tff(fact_7029_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,aa(nat,fun(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_7030_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_7031_sint__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(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_7032_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),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),Xa)),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_7033_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),I)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),J)) ) ) ) ).

% word_add_le_mono2
tff(fact_7034_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_le_mono1
tff(fact_7035_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_7036_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_7037_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(nat,$o,aa(nat,fun(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),Xa)),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_7038_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_7039_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_dest
tff(fact_7040_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_iff
tff(fact_7041_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(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,Xa)) ) ).

% sint_ge
tff(fact_7042_unat__less__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,K: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(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_7043_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xa: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ! [I3: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I3) = Xa )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(int,$o,P,I3) ) ) ) ).

% uint_split
tff(fact_7044_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xa: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ~ ? [I3: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I3) = Xa )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(int,$o,P,I3) ) ) ) ).

% uint_split_asm
tff(fact_7045_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(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_7046_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_mult_less_dest
tff(fact_7047_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(nat,$o,aa(nat,fun(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),Xa))),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_7048_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_7049_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(nat,$o,aa(nat,fun(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),Xa))),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_7050_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ? [Z3: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z3) )
              & aa(int,$o,aa(int,fun(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),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Z3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_le_exists'
tff(fact_7051_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa))
        <=> aa(int,$o,aa(int,fun(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),Xa))),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_7052_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(int,$o,aa(int,fun(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),Xa)),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_7053_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: nat,Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(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))),Xa)) ) ) ) ).

% More_Word.of_nat_power
tff(fact_7054_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,aa(int,fun(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_7055_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Nb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_7056_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Nb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_7057_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_sub_if'
tff(fact_7058_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),int,semiring_1_unsigned(A,int),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2))) = modulo_modulo(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_7059_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ) ).

% word_less_two_pow_divI
tff(fact_7060_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
           => ( ( Xa != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(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_7061_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(int,$o,aa(int,fun(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),Xa))),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_7062_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),bit_se2239418461657761734s_mask(word(A),M))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),bit_se2239418461657761734s_mask(word(A),M))
             => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(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)) )
               => ( Xa = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_7063_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(int,$o,aa(int,fun(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),Xa))),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_7064_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(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_7065_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(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_7066_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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)),Xa)) = aa(nat,int,semiring_1_of_nat(int),Xa) ) ) ) ).

% int_eq_sint
tff(fact_7067_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ) ).

% word_mult_less_cancel
tff(fact_7068_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_7069_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(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_7070_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
           => ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N4)
               => ( aa(nat,$o,aa(nat,fun(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_7071_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N4)
             => ( aa(nat,$o,aa(nat,fun(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_7072_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: nat,Xa: word(A),M: nat,Sz: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) )
                 => aa(word(A),$o,aa(word(A),fun(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)),Xa),aa(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_7073_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)) = 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),Xa))),one_one(word(A))) ) ) ) ) ).

% div_power_helper
tff(fact_7074_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),M: nat,Nb: nat] :
          ( aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)))
        <=> ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
              & ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_word_iff
tff(fact_7075_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(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_7076_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: nat,A2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),A2))),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),B2))) ) ) ) ).

% sint_of_nat_le
tff(fact_7077_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),Xa))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_7078_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))))) = 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_7079_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),uminus_uminus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),Xa)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),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)),Xa)) = Xa ) ) ) ) ).

% sint_of_int_eq
tff(fact_7080_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_7081_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_7082_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_le_mono1
tff(fact_7083_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),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_7084_int__word__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Xa)) = aa(int,int,minus_minus(int,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),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_7085_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
         => ( aa(nat,$o,aa(nat,fun(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),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Y)),Y) = Xa ) ) ) ) ).

% Word.word_div_mult
tff(fact_7086_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
         => ( aa(fun(word(A),$o),set(word(A)),collect(word(A)),aa(nat,fun(word(A),$o),aTP_Lamp_yt(nat,fun(nat,fun(word(A),$o)),Nb),M)) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_yu(nat,fun(nat,fun(nat,$o)),Nb),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_7087_sint__int__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,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)))))) = 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_7088_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),A2: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [A4: word(A)] :
                ( aa(word(A),$o,P,A4)
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),A4)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A4),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4)) ) ) )
           => ( ! [A4: word(A)] :
                  ( aa(word(A),$o,P,A4)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A4),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4))) ) )
             => aa(word(A),$o,P,A2) ) ) ) ) ).

% word_bit_induct
tff(fact_7089_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(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_7090_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(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_7091_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: $o] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa($o,word(A),zero_neq_one_of_bool(word(A)),(B2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = A2 ) ) ) ).

% bit_word_half_eq
tff(fact_7092_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,aa(list(word(A)),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,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),type_len0_len_of(A,type2(A)))),aa(list(word(A)),nat,size_size(list(word(A))),Ws))
            & aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(nat,word(A),nth(word(A),Ws),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),type_len0_len_of(A,type2(A))))),modulo_modulo(nat,Nb,type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_7093_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 = 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,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,i),Shift)),Least)
                            | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Overflow),aa(int,int,minus_minus(int,i),Shift)) ),
                            aa(int,word(A),Arbitrary1,i),
                            aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,i),Shift)) ),
                          $ite(
                            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),i),Least)
                            | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Overflow),i) ),
                            aa(int,word(A),Arbitrary2,i),
                            aa(int,word(A),ring_1_of_int(word(A)),i) ) ) ) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
tff(fact_7094_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,Us: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Us),Sz)
           => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Sz),Us)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Us)))) ) ) ) ) ).

% Suc_div_unat_helper
tff(fact_7095_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Nb: nat,Xa: word(A),A2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)))
         => ( ( Xa = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),Xa)
             => ( aa(nat,$o,aa(nat,fun(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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb))),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) ) ) ) ) ) ) ).

% alignUp_div_helper
tff(fact_7096_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A))) ) ) ).

% less_eq_decr_length_iff
tff(fact_7097_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Nb)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb) ) ) ).

% decr_length_less_iff
tff(fact_7098_len__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),type_len0_len_of(A,type2(A))) ) ).

% len_gt_0
tff(fact_7099_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% length_not_greater_eq_2_iff
tff(fact_7100_len__num0,axiom,
    ! [Uu: itself(numeral_num0)] : type_len0_len_of(numeral_num0,Uu) = zero_zero(nat) ).

% len_num0
tff(fact_7101_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_7102_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_7103_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% two_less_eq_exp_length
tff(fact_7104_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_7105_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),modulo_modulo(word(A),Xa,Y)) = $ite(
                aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),zero_zero(word(A))),Xa),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),one_one(word(A))),aa(word(A),word(A),minus_minus(word(A),Xa),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),Xa),Y)),
                  $let(
                    r: word(A),
                    r:= aa(word(A),word(A),minus_minus(word(A),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                    $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),r),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),q),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),r),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),q),r)) ) ) ) ) ) ) ).

% divmod_via_sdivmod
tff(fact_7106_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),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_minus_numeral
tff(fact_7107_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_7108_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_7109_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_7110_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_7111_signed__drop__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : signed_drop_bit(A,Nb,uminus_uminus(word(A),one_one(word(A)))) = uminus_uminus(word(A),one_one(word(A))) ) ).

% signed_drop_bit_of_minus_1
tff(fact_7112_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_7113_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_7114_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_7115_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),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_Suc_minus_numeral
tff(fact_7116_sint__signed__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] : ring_1_signed(A,int,signed_drop_bit(A,Nb,W)) = bit_se4197421643247451524op_bit(int,Nb,ring_1_signed(A,int,W)) ) ).

% sint_signed_drop_bit_eq
tff(fact_7117_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_7118_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_7119_word__sdiv__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Xa),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)),Xa)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_sdiv_numerals(4)
tff(fact_7120_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_7121_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_7122_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_7123_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Nb)
                & aa(nat,$o,aa(nat,fun(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_7124_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( signed_drop_bit(A,Nb,W) = $ite(aa(nat,$o,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)))),uminus_uminus(word(A),one_one(word(A))),zero_zero(word(A))) ) ) ) ).

% signed_drop_bit_beyond
tff(fact_7125_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F: fun(int,A),Xa: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F,Xa))
        <=> ! [I3: int] :
              ( ( ( Xa = aa(int,word(B),ring_1_of_int(word(B)),I3) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B)))) )
             => aa(A,$o,P,aa(int,A,F,I3)) ) ) ) ).

% word_int_split
tff(fact_7126_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F: fun(int,A),Xa: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F,Xa))
        <=> ~ ? [N6: int] :
                ( ( Xa = aa(int,word(B),ring_1_of_int(word(B)),N6) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N6)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),N6),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))
                & ~ aa(A,$o,P,aa(int,A,F,N6)) ) ) ) ).

% word_int_split_asm
tff(fact_7127_word__int__case__eq__uint,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F: fun(int,A),W: word(B)] : word_int_case(A,B,F,W) = aa(int,A,F,aa(word(B),int,semiring_1_unsigned(B,int),W)) ) ).

% word_int_case_eq_uint
tff(fact_7128_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F: fun(int,A),I: int] : word_int_case(A,B,F,aa(int,word(B),ring_1_of_int(word(B)),I)) = aa(int,A,F,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_7129_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_7130_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat,W: word(B)] :
          aa(word(B),word(A),slice1(B,A,Nb),W) = $ite(aa(nat,$o,aa(nat,fun(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_7131_slice1__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat] : aa(word(B),word(A),slice1(B,A,Nb),zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice1_0
tff(fact_7132_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_7133_succ__pred__no_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_pred(A,uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),W))) = aa(word(A),word(A),minus_minus(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_7134_word__m1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),word_pred(A,zero_zero(word(A)))) ) ).

% word_m1_ge
tff(fact_7135_word__not__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),word_pred(A,zero_zero(word(A)))),Y) ) ).

% word_not_simps(2)
tff(fact_7136_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_7137_mask__eqs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_pred(A,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_pred(A,A2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(12)
tff(fact_7138_ucast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = aa(word(B),word(A),slice1(B,A,aa(word(B),nat,size_size(word(B)),W)),W) ) ).

% ucast_slice1
tff(fact_7139_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_7140_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)),uminus_uminus(int,one_one(int))) ) ) ).

% word_pred_0_n1
tff(fact_7141_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_7142_Word_Oslice__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat] : slice2(A,B,Nb) = slice1(A,B,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)) ) ).

% Word.slice_def
tff(fact_7143_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_7144_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_7145_word__reverse__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : word_reverse(A,W) = aa(list($o),word(A),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(list($o),list($o),rev($o),map(nat,$o,bit_se5641148757651400278ts_bit(word(A),W),upt(zero_zero(nat),type_len0_len_of(A,type2(A)))))) ) ).

% word_reverse_def
tff(fact_7146_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_7147_word__rev__rev,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : word_reverse(A,word_reverse(A,W)) = W ) ).

% word_rev_rev
tff(fact_7148_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_7149_succ__pred__no_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_succ(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)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),W))),one_one(word(A))) ) ).

% succ_pred_no(3)
tff(fact_7150_word__pred__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_pred(A,word_succ(A,A2)) = A2 ) ).

% word_pred_succ
tff(fact_7151_word__succ__pred,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,word_pred(A,A2)) = A2 ) ).

% word_succ_pred
tff(fact_7152_word__rev__gal_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),W: word(A)] :
          ( ( U = word_reverse(A,W) )
         => ( W = word_reverse(A,U) ) ) ) ).

% word_rev_gal'
tff(fact_7153_word__rev__gal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),U: word(A)] :
          ( ( word_reverse(A,W) = U )
         => ( word_reverse(A,U) = W ) ) ) ).

% word_rev_gal
tff(fact_7154_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_7155_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_7156_mask__eqs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_succ(A,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_succ(A,A2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(11)
tff(fact_7157_word__mult__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),word_succ(A,A2)),B2) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) ) ).

% word_mult_succ
tff(fact_7158_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_7159_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_7160_rev__slice,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat,K: nat,W: word(B)] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)),type_len0_len_of(A,type2(A))) = type_len0_len_of(B,type2(B)) )
         => ( aa(word(B),word(A),slice2(B,A,Nb),word_reverse(B,W)) = word_reverse(A,aa(word(B),word(A),slice2(B,A,K),W)) ) ) ) ).

% rev_slice
tff(fact_7161_rev__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat,K: nat,W: word(B)] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K) = 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))) )
         => ( aa(word(B),word(A),slice1(B,A,Nb),word_reverse(B,W)) = word_reverse(A,aa(word(B),word(A),slice1(B,A,K),W)) ) ) ) ).

% rev_slice1
tff(fact_7162_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_7163_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,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))) = uminus_uminus(word(C),one_one(word(C))) )
        & ( word_pred(D6,one_one(word(D6))) = zero_zero(word(D6)) ) ) ) ).

% word_sp_01
tff(fact_7164_bit__word__reverse__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),word_reverse(A,W)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(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,Nb))) ) ) ) ).

% bit_word_reverse_iff
tff(fact_7165_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_7166_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_7167_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_7168_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(list($o),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,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(list($o),nat,size_size(list($o)),Bs)))
            & aa(nat,$o,nth($o,Bs),Nb) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_7169_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,$o,aa(nat,fun(nat,$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_7170_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb) = M ) ) ) ).

% min_arg_le(2)
tff(fact_7171_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Nb: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb) = Nb ) ) ) ).

% min_arg_le(1)
tff(fact_7172_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb) = Nb )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),M) ) ) ).

% min_eq_arg(2)
tff(fact_7173_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb) = M )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),Nb) ) ) ).

% min_eq_arg(1)
tff(fact_7174_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% min.bounded_iff
tff(fact_7175_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_7176_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_7177_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min_simps(2)
tff(fact_7178_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min_simps(1)
tff(fact_7179_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% min_less_self_conv(2)
tff(fact_7180_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% min_less_self_conv(1)
tff(fact_7181_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb)),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb) = Nb ) ) ) ).

% min_arg_not_ge(2)
tff(fact_7182_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb)),M)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Nb) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_7183_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_7184_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_7185_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_7186_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_7187_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% min_bot
tff(fact_7188_min__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Nb)) ).

% min_Suc_Suc
tff(fact_7189_min__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% min_0L
tff(fact_7190_min__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_7191_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_7192_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_7193_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_7194_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_7195_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),Va2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),Va2)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),Va2)) ) ).

% min_number_of(1)
tff(fact_7196_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).

% min_0_1(1)
tff(fact_7197_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(2)
tff(fact_7198_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A2)),B2) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_7199_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),aa(nat,nat,suc,A2)) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_7200_length__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),take(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_take
tff(fact_7201_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),uminus_uminus(A,aa(num,A,numeral_numeral(A),U)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) ) ).

% min_number_of(4)
tff(fact_7202_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),Va2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),Va2)),uminus_uminus(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),Va2)) ) ).

% min_number_of(3)
tff(fact_7203_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,Va2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))),aa(num,A,numeral_numeral(A),U),uminus_uminus(A,aa(num,A,numeral_numeral(A),Va2))) ) ).

% min_number_of(2)
tff(fact_7204_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_7205_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),Nb)) ).

% min_numeral_Suc
tff(fact_7206_length__zip,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))),zip(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_zip
tff(fact_7207_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,aa(nat,nat,aa(nat,fun(nat,nat),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_7208_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,aa(nat,nat,aa(nat,fun(nat,nat),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_7209_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)),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,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,Nb))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_Suc_minus_numeral
tff(fact_7210_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)),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,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(num,nat,numeral_numeral(nat),Nb))),uminus_uminus(int,aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_minus_numeral
tff(fact_7211_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_min(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% min_add_distrib_right
tff(fact_7212_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_7213_word__roti__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: int,Nb: int,W: word(A)] : word_roti(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),Nb),W) = word_roti(A,M,word_roti(A,Nb,W)) ) ).

% word_roti_add
tff(fact_7214_nat__mult__min__left,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Nb)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ).

% nat_mult_min_left
tff(fact_7215_nat__mult__min__right,axiom,
    ! [M: nat,Nb: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),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),Q4)) ).

% nat_mult_min_right
tff(fact_7216_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : uminus_uminus(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),uminus_uminus(A,Xa)),uminus_uminus(A,Y)) ) ).

% minus_min_eq_max
tff(fact_7217_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : uminus_uminus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),uminus_uminus(A,Xa)),uminus_uminus(A,Y)) ) ).

% minus_max_eq_min
tff(fact_7218_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),A2,B2) ) ).

% min_def
tff(fact_7219_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Xa ) ) ) ).

% min_absorb1
tff(fact_7220_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_7221_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D)) ) ) ) ).

% min.mono
tff(fact_7222_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_7223_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% min.orderI
tff(fact_7224_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% min.boundedE
tff(fact_7225_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ) ) ).

% min.boundedI
tff(fact_7226_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.order_iff
tff(fact_7227_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2) ) ).

% min.cobounded1
tff(fact_7228_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2) ) ).

% min.cobounded2
tff(fact_7229_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_7230_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_7231_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.coboundedI1
tff(fact_7232_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.coboundedI2
tff(fact_7233_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_7234_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X4),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa2),X4,Xa2) ) ).

% min_def_raw
tff(fact_7235_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_7236_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_7237_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_7238_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_7239_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_7240_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_7241_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_7242_min__diff,axiom,
    ! [M: nat,I: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,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,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Nb)),I) ).

% min_diff
tff(fact_7243_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,minus_minus(A,Xa),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_7244_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(product_prod(A,B)),$o)] :
      ( ! [Zs2: list(A),Ws2: list(B),N: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws2) )
         => ( ( N = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) )
           => ( ( Zs2 = take(A,N,Xs) )
             => ( ( Ws2 = take(B,N,Ys) )
               => aa(list(product_prod(A,B)),$o,P,zip(A,B,Zs2,Ws2)) ) ) ) )
     => aa(list(product_prod(A,B)),$o,P,zip(A,B,Xs,Ys)) ) ).

% zip_obtain_same_length
tff(fact_7245_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% min_mult_distrib_right
tff(fact_7246_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% max_mult_distrib_right
tff(fact_7247_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xa: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% min_mult_distrib_left
tff(fact_7248_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xa: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% max_mult_distrib_left
tff(fact_7249_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3))) ) ).

% min_divide_distrib_right
tff(fact_7250_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3))) ) ).

% max_divide_distrib_right
tff(fact_7251_min__Suc1,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_yv(nat,fun(nat,nat),Nb),M) ).

% min_Suc1
tff(fact_7252_min__Suc2,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_yw(nat,fun(nat,nat),Nb),M) ).

% min_Suc2
tff(fact_7253_ucast__mask__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),bit_se2239418461657761734s_mask(word(B),Nb)) = bit_se2239418461657761734s_mask(word(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(B,type2(B))),Nb)) ) ).

% ucast_mask_eq
tff(fact_7254_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),M)) ) ) ).

% Word.bit_mask_iff
tff(fact_7255_uint__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : aa(word(A),int,semiring_1_unsigned(A,int),bit_se2239418461657761734s_mask(word(A),Nb)) = bit_se2239418461657761734s_mask(int,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),Nb)) ) ).

% uint_mask_eq
tff(fact_7256_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_7257_and__mask__wi_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),ring_1_of_int(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,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),Nb)),I)) ) ).

% and_mask_wi'
tff(fact_7258_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_7259_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),aa(word(B),word(A),slice2(B,A,M),W)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),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_7260_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),aa(word(B),word(A),slice1(B,A,M),W)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),type_len0_len_of(B,type2(B)))),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),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_7261_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),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_7262_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_7263_signed_Odual__order_Orefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),A2) ) ).

% signed.dual_order.refl
tff(fact_7264_signed_Oorder__refl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Xa) ) ).

% signed.order_refl
tff(fact_7265_min__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),Q4),zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_7266_min__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),zero_zero(extended_enat)),Q4) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_7267_signed_OMin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3))
            <=> ! [X3: word(A)] :
                  ( aa(set(word(A)),$o,member(word(A),X3),A3)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),X3) ) ) ) ) ) ).

% signed.Min.bounded_iff
tff(fact_7268_signed_OMin__singleton,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),aa(set(word(A)),set(word(A)),insert(word(A),Xa),bot_bot(set(word(A))))) = Xa ) ).

% signed.Min_singleton
tff(fact_7269_word__sle__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),A2))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_sle_no
tff(fact_7270_signed_OMin__const,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [A3: set(A),C2: word(B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(word(B)),word(B),lattices_Min(word(B),word_sle(B)),aa(set(A),set(word(B)),image(A,word(B),aTP_Lamp_yx(word(B),fun(A,word(B)),C2)),A3)) = C2 ) ) ) ) ).

% signed.Min_const
tff(fact_7271_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))) ) ) ).

% extra_sle_sless_unfolds(4)
tff(fact_7272_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Nb)),one_one(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(6)
tff(fact_7273_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Nb)),zero_zero(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(5)
tff(fact_7274_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))) ) ) ).

% extra_sle_sless_unfolds(2)
tff(fact_7275_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),one_one(word(A))),zero_zero(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(3)
tff(fact_7276_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),one_one(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(1)
tff(fact_7277_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_7278_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_7279_signed_Osorted__replicate,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A)] : sorted_wrt(word(A),word_sle(A),replicate(word(A),Nb,Xa)) ) ).

% signed.sorted_replicate
tff(fact_7280_signed_OMin__insert2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),A2: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ! [B3: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),B3),A3)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B3) )
           => ( aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),aa(set(word(A)),set(word(A)),insert(word(A),A2),A3)) = A2 ) ) ) ) ).

% signed.Min_insert2
tff(fact_7281_signed_Ofinite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ? [X: list(word(A))] :
              ( ( aa(list(word(A)),set(word(A)),set2(word(A)),X) = A3 )
              & sorted_wrt(word(A),word_sle(A),X)
              & distinct(word(A),X)
              & ! [Y4: list(word(A))] :
                  ( ( ( aa(list(word(A)),set(word(A)),set2(word(A)),Y4) = A3 )
                    & sorted_wrt(word(A),word_sle(A),Y4)
                    & distinct(word(A),Y4) )
                 => ( Y4 = X ) ) ) ) ) ).

% signed.finite_sorted_distinct_unique
tff(fact_7282_signed_Osorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),Ys: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => ( distinct(word(A),Xs)
           => ( sorted_wrt(word(A),word_sle(A),Ys)
             => ( distinct(word(A),Ys)
               => ( ( aa(list(word(A)),set(word(A)),set2(word(A)),Xs) = aa(list(word(A)),set(word(A)),set2(word(A)),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% signed.sorted_distinct_set_unique
tff(fact_7283_signed_Osorted__drop,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),Nb: nat] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => sorted_wrt(word(A),word_sle(A),drop(word(A),Nb,Xs)) ) ) ).

% signed.sorted_drop
tff(fact_7284_signed_Osorted__append,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),Ys: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),append(word(A),Xs,Ys))
        <=> ( sorted_wrt(word(A),word_sle(A),Xs)
            & sorted_wrt(word(A),word_sle(A),Ys)
            & ! [X3: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X3),aa(list(word(A)),set(word(A)),set2(word(A)),Xs))
               => ! [Xa3: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),Xa3),aa(list(word(A)),set(word(A)),set2(word(A)),Ys))
                   => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X3),Xa3) ) ) ) ) ) ).

% signed.sorted_append
tff(fact_7285_signed_Osorted__take,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),Nb: nat] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => sorted_wrt(word(A),word_sle(A),take(word(A),Nb,Xs)) ) ) ).

% signed.sorted_take
tff(fact_7286_signed_Odual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% signed.dual_order.antisym
tff(fact_7287_signed_Odual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( A2 = B2 )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2) ) ) ) ).

% signed.dual_order.eq_iff
tff(fact_7288_signed_Odual__order_Otrans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),C2),A2) ) ) ) ).

% signed.dual_order.trans
tff(fact_7289_signed_Oord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),C2) ) ) ) ).

% signed.ord_le_eq_trans
tff(fact_7290_signed_Oord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( ( A2 = B2 )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),C2) ) ) ) ).

% signed.ord_eq_le_trans
tff(fact_7291_signed_Oorder__antisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa)
           => ( Xa = Y ) ) ) ) ).

% signed.order_antisym
tff(fact_7292_signed_Oorder_Oantisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% signed.order.antisym
tff(fact_7293_signed_Olinorder__wlog,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),fun(word(A),$o)),A2: word(A),B2: word(A)] :
          ( ! [A4: word(A),B3: word(A)] :
              ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A4),B3)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B3) )
         => ( ! [A4: word(A),B3: word(A)] :
                ( aa(word(A),$o,aa(word(A),fun(word(A),$o),P,B3),A4)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B3) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A2),B2) ) ) ) ).

% signed.linorder_wlog
tff(fact_7294_signed_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa) ) ) ) ).

% signed.order_eq_iff
tff(fact_7295_signed_Oorder_Oeq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( A2 = B2 )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2) ) ) ) ).

% signed.order.eq_iff
tff(fact_7296_signed_Oantisym__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% signed.antisym_conv
tff(fact_7297_signed_Oorder__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Z) ) ) ) ).

% signed.order_trans
tff(fact_7298_signed_Oorder_Otrans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),C2) ) ) ) ).

% signed.order.trans
tff(fact_7299_signed_Ole__cases3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
           => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z) )
         => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa)
             => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Z) )
           => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Z)
               => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),Y) )
             => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),Y)
                 => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa) )
               => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z)
                   => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),Xa) )
                 => ~ ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),Xa)
                     => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y) ) ) ) ) ) ) ) ).

% signed.le_cases3
tff(fact_7300_signed_Ole__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa) ) ) ).

% signed.le_cases
tff(fact_7301_signed_Oeq__refl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y) ) ) ).

% signed.eq_refl
tff(fact_7302_signed_Onle__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% signed.nle_le
tff(fact_7303_signed_Olinear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
          | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa) ) ) ).

% signed.linear
tff(fact_7304_signed_Osorted__remove1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),A2: word(A)] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => sorted_wrt(word(A),word_sle(A),remove1(word(A),A2,Xs)) ) ) ).

% signed.sorted_remove1
tff(fact_7305_signed_Osorted__tl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),tl(word(A)),Xs)) ) ) ).

% signed.sorted_tl
tff(fact_7306_signed_Osorted__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [G: fun(list(word(A)),word(A)),Xs: list(word(A))] : sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),filter2(word(A),aa(list(word(A)),fun(word(A),$o),aTP_Lamp_yy(fun(list(word(A)),word(A)),fun(list(word(A)),fun(word(A),$o)),G),Xs)),Xs)) ) ).

% signed.sorted_same
tff(fact_7307_signed_Osorted__takeWhile,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),P: fun(word(A),$o)] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => sorted_wrt(word(A),word_sle(A),takeWhile(word(A),P,Xs)) ) ) ).

% signed.sorted_takeWhile
tff(fact_7308_signed_Osorted0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => sorted_wrt(word(A),word_sle(A),nil(word(A))) ) ).

% signed.sorted0
tff(fact_7309_signed_OMin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),A2: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),$o,member(word(A),A2),A3)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3)),A2) ) ) ) ).

% signed.Min.coboundedI
tff(fact_7310_signed_OMin__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ! [Y3: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),Y3),A3)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y3) )
           => ( aa(set(word(A)),$o,member(word(A),Xa),A3)
             => ( aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3) = Xa ) ) ) ) ) ).

% signed.Min_eqI
tff(fact_7311_signed_OMin__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),$o,member(word(A),Xa),A3)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3)),Xa) ) ) ) ).

% signed.Min_le
tff(fact_7312_signed_Ofinite__has__maximal2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),A2: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),$o,member(word(A),A2),A3)
           => ? [X: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X),A3)
                & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),X)
                & ! [Xa2: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),Xa2),A3)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% signed.finite_has_maximal2
tff(fact_7313_signed_Ofinite__has__minimal2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),A2: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),$o,member(word(A),A2),A3)
           => ? [X: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X),A3)
                & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),A2)
                & ! [Xa2: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),Xa2),A3)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% signed.finite_has_minimal2
tff(fact_7314_signed_Osorted__butlast,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( ( Xs != nil(word(A)) )
         => ( sorted_wrt(word(A),word_sle(A),Xs)
           => sorted_wrt(word(A),word_sle(A),butlast(word(A),Xs)) ) ) ) ).

% signed.sorted_butlast
tff(fact_7315_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F,aa(nat,nat,suc,N))),aa(nat,word(A),F,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F,N3)),aa(nat,word(A),F,Nb)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_7316_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F,N)),aa(nat,word(A),F,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F,Nb)),aa(nat,word(A),F,N3)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_7317_word__sle__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2)) ) ) ).

% word_sle_eq
tff(fact_7318_signed_OMin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),B4: set(word(A))] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),A3),B4)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),B4)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),B4)),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3)) ) ) ) ) ).

% signed.Min.subset_imp
tff(fact_7319_signed_OMin__antimono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M6: set(word(A)),N7: set(word(A))] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),M6),N7)
         => ( ( M6 != bot_bot(set(word(A))) )
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),N7)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),N7)),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),M6)) ) ) ) ) ).

% signed.Min_antimono
tff(fact_7320_signed_OMin__in,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => aa(set(word(A)),$o,member(word(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3)),A3) ) ) ) ).

% signed.Min_in
tff(fact_7321_signed_OMin__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),M: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( ( aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3) = M )
            <=> ( aa(set(word(A)),$o,member(word(A),M),A3)
                & ! [X3: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),X3),A3)
                   => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),M),X3) ) ) ) ) ) ) ).

% signed.Min_eq_iff
tff(fact_7322_signed_OMin__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3)),Xa)
            <=> ? [X3: word(A)] :
                  ( aa(set(word(A)),$o,member(word(A),X3),A3)
                  & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X3),Xa) ) ) ) ) ) ).

% signed.Min_le_iff
tff(fact_7323_signed_Oeq__Min__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),M: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( ( M = aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3) )
            <=> ( aa(set(word(A)),$o,member(word(A),M),A3)
                & ! [X3: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),X3),A3)
                   => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),M),X3) ) ) ) ) ) ) ).

% signed.eq_Min_iff
tff(fact_7324_signed_OMin_OboundedE,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3))
             => ! [A11: word(A)] :
                  ( aa(set(word(A)),$o,member(word(A),A11),A3)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),A11) ) ) ) ) ) ).

% signed.Min.boundedE
tff(fact_7325_signed_OMin_OboundedI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( ! [A4: word(A)] :
                  ( aa(set(word(A)),$o,member(word(A),A4),A3)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),A4) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3)) ) ) ) ) ).

% signed.Min.boundedI
tff(fact_7326_signed_Ofinite__has__maximal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ? [X: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X),A3)
                & ! [Xa2: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),Xa2),A3)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% signed.finite_has_maximal
tff(fact_7327_signed_Ofinite__has__minimal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ? [X: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X),A3)
                & ! [Xa2: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),Xa2),A3)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% signed.finite_has_minimal
tff(fact_7328_signed_Osorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xs: list(B),P: fun(B,$o)] :
          ( sorted_wrt(word(A),word_sle(A),map(B,word(A),F,Xs))
         => sorted_wrt(word(A),word_sle(A),map(B,word(A),F,aa(list(B),list(B),filter2(B,P),Xs))) ) ) ).

% signed.sorted_filter
tff(fact_7329_signed_Osorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xs: list(B),Xa: B] :
          ( sorted_wrt(word(A),word_sle(A),map(B,word(A),F,Xs))
         => sorted_wrt(word(A),word_sle(A),map(B,word(A),F,remove1(B,Xa,Xs))) ) ) ).

% signed.sorted_map_remove1
tff(fact_7330_signed_Osorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),G: fun(list(B),word(A)),Xs: list(B)] : sorted_wrt(word(A),word_sle(A),map(B,word(A),F,aa(list(B),list(B),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),word(A)),fun(list(B),fun(B,$o)),aTP_Lamp_yz(fun(B,word(A)),fun(fun(list(B),word(A)),fun(list(B),fun(B,$o))),F),G),Xs)),Xs))) ) ).

% signed.sorted_map_same
tff(fact_7331_signed_Osorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xs: list(B)] :
          ( sorted_wrt(word(A),word_sle(A),map(B,word(A),F,Xs))
        <=> sorted_wrt(B,aTP_Lamp_za(fun(B,word(A)),fun(B,fun(B,$o)),F),Xs) ) ) ).

% signed.sorted_map
tff(fact_7332_signed_OMin_Oinfinite,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( ~ aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3) = aa(option(word(A)),word(A),the2(word(A)),none(word(A))) ) ) ) ).

% signed.Min.infinite
tff(fact_7333_signed_Osorted2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Zs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),cons(word(A),Xa),aa(list(word(A)),list(word(A)),cons(word(A),Y),Zs)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
            & sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),cons(word(A),Y),Zs)) ) ) ) ).

% signed.sorted2
tff(fact_7334_signed_Osorted1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),cons(word(A),Xa),nil(word(A)))) ) ).

% signed.sorted1
tff(fact_7335_signed_Osorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ys: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),cons(word(A),Xa),Ys))
        <=> ( ! [X3: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X3),aa(list(word(A)),set(word(A)),set2(word(A)),Ys))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),X3) )
            & sorted_wrt(word(A),word_sle(A),Ys) ) ) ) ).

% signed.sorted_simps(2)
tff(fact_7336_signed_Ofinite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [S3: set(A),P: fun(set(A),$o),F: fun(A,word(B))] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X: A,S5: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S5)
                 => ( ! [Y4: A] :
                        ( aa(set(A),$o,member(A,Y4),S5)
                       => aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F,Y4)),aa(A,word(B),F,X)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X),S5)) ) ) )
             => aa(set(A),$o,P,S3) ) ) ) ) ).

% signed.finite_ranking_induct
tff(fact_7337_signed_Osorted01,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(word(A)),nat,size_size(list(word(A))),Xs)),one_one(nat))
         => sorted_wrt(word(A),word_sle(A),Xs) ) ) ).

% signed.sorted01
tff(fact_7338_signed_Osorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I3)),aa(nat,word(A),nth(word(A),Xs),J3)) ) ) ) ) ).

% signed.sorted_iff_nth_mono_less
tff(fact_7339_signed_Osorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I3)),aa(nat,word(A),nth(word(A),Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% signed.sorted_iff_nth_Suc
tff(fact_7340_signed_Osorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I3)),aa(nat,word(A),nth(word(A),Xs),J3)) ) ) ) ) ).

% signed.sorted_iff_nth_mono
tff(fact_7341_signed_Osorted__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),I: nat,J: nat] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I)),aa(nat,word(A),nth(word(A),Xs),J)) ) ) ) ) ).

% signed.sorted_nth_mono
tff(fact_7342_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: 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)),Xa),Y)),
              $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),uminus_uminus(word(A),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_min(word(A)),Xa),Y) ) ).

% bit_twiddle_min
tff(fact_7343_signed_Osorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),rev(word(A)),Xs))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),aa(nat,nat,suc,I3))),aa(nat,word(A),nth(word(A),Xs),I3)) ) ) ) ).

% signed.sorted_rev_iff_nth_Suc
tff(fact_7344_signed_Osorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),I: nat,J: nat] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),rev(word(A)),Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),J)),aa(nat,word(A),nth(word(A),Xs),I)) ) ) ) ) ).

% signed.sorted_rev_nth_mono
tff(fact_7345_signed_Osorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),rev(word(A)),Xs))
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),J3)),aa(nat,word(A),nth(word(A),Xs),I3)) ) ) ) ) ).

% signed.sorted_rev_iff_nth_mono
tff(fact_7346_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),Xa) ) ) ).

% word_0_sle_from_less
tff(fact_7347_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(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))),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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_7348_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(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))),uminus_uminus(int,aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_7349_word__sless__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
            & ( aa(num,word(A),numeral_numeral(word(A)),A2) != aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ) ).

% word_sless_no
tff(fact_7350_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Nb)),zero_zero(word(A)))
        <=> aa(int,$o,aa(int,fun(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_7351_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Nb))
        <=> aa(int,$o,aa(int,fun(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_7352_extra__sle__sless__unfolds_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),zero_zero(word(A))),one_one(word(A)))
      <=> aa(int,$o,aa(int,fun(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_7353_extra__sle__sless__unfolds_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),one_one(word(A))),zero_zero(word(A)))
      <=> aa(int,$o,aa(int,fun(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_7354_signed_OMin__gr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3))
            <=> ! [X3: word(A)] :
                  ( aa(set(word(A)),$o,member(word(A),X3),A3)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),X3) ) ) ) ) ) ).

% signed.Min_gr_iff
tff(fact_7355_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(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_7356_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),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_7357_word__sless__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
            & ( Xa != Y ) ) ) ) ).

% word_sless_eq
tff(fact_7358_signed_Odual__order_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( A2 != B2 )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2) ) ) ) ).

% signed.dual_order.not_eq_order_implies_strict
tff(fact_7359_signed_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( A2 != B2 )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2) ) ) ) ).

% signed.order.not_eq_order_implies_strict
tff(fact_7360_signed_Odual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2) ) ) ).

% signed.dual_order.strict_implies_order
tff(fact_7361_signed_Odual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% signed.dual_order.strict_iff_order
tff(fact_7362_signed_Odual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% signed.dual_order.order_iff_strict
tff(fact_7363_signed_Oorder_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2) ) ) ).

% signed.order.strict_implies_order
tff(fact_7364_signed_Odual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
            & ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2) ) ) ) ).

% signed.dual_order.strict_iff_not
tff(fact_7365_signed_Odual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),A2) ) ) ) ).

% signed.dual_order.strict_trans2
tff(fact_7366_signed_Odual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),A2) ) ) ) ).

% signed.dual_order.strict_trans1
tff(fact_7367_signed_Oorder_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% signed.order.strict_iff_order
tff(fact_7368_signed_Oorder_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% signed.order.order_iff_strict
tff(fact_7369_signed_Oorder_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
            & ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),A2) ) ) ) ).

% signed.order.strict_iff_not
tff(fact_7370_signed_Oorder_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),B2),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),C2) ) ) ) ).

% signed.order.strict_trans2
tff(fact_7371_signed_Oorder_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),C2) ) ) ) ).

% signed.order.strict_trans1
tff(fact_7372_signed_Ole__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% signed.le_imp_less_or_eq
tff(fact_7373_signed_Onot__le__imp__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y) ) ) ).

% signed.not_le_imp_less
tff(fact_7374_signed_Oless__le__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
            & ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa) ) ) ) ).

% signed.less_le_not_le
tff(fact_7375_signed_Ole__less__linear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
          | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ).

% signed.le_less_linear
tff(fact_7376_signed_Oless__le__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Z) ) ) ) ).

% signed.less_le_trans
tff(fact_7377_signed_Ole__less__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Z) ) ) ) ).

% signed.le_less_trans
tff(fact_7378_signed_Oantisym__conv2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
         => ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% signed.antisym_conv2
tff(fact_7379_signed_Oantisym__conv1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% signed.antisym_conv1
tff(fact_7380_signed_Ole__neq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
         => ( ( A2 != B2 )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2) ) ) ) ).

% signed.le_neq_trans
tff(fact_7381_signed_Oless__imp__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y) ) ) ).

% signed.less_imp_le
tff(fact_7382_signed_Onot__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa) ) ) ).

% signed.not_less
tff(fact_7383_signed_Onless__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
        <=> ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% signed.nless_le
tff(fact_7384_signed_Ole__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% signed.le_less
tff(fact_7385_signed_Onot__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ).

% signed.not_le
tff(fact_7386_signed_OleI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa) ) ) ).

% signed.leI
tff(fact_7387_signed_OleD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Xa)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y) ) ) ).

% signed.leD
tff(fact_7388_signed_Ostrict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),Xs)
         => sorted_wrt(word(A),word_sle(A),Xs) ) ) ).

% signed.strict_sorted_imp_sorted
tff(fact_7389_signed_Ostrict__sorted__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [L: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),L)
        <=> ( sorted_wrt(word(A),word_sle(A),L)
            & distinct(word(A),L) ) ) ) ).

% signed.strict_sorted_iff
tff(fact_7390_signed_OMin__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A3)),Xa)
            <=> ? [X3: word(A)] :
                  ( aa(set(word(A)),$o,member(word(A),X3),A3)
                  & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X3),Xa) ) ) ) ) ) ).

% signed.Min_less_iff
tff(fact_7391_signed_Ostrict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ys: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),aa(list(word(A)),list(word(A)),cons(word(A),Xa),Ys))
        <=> ( ! [X3: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X3),aa(list(word(A)),set(word(A)),set2(word(A)),Ys))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),X3) )
            & sorted_wrt(word(A),word_sless(A),Ys) ) ) ) ).

% signed.strict_sorted_simps(2)
tff(fact_7392_signed_Ofinite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),P: fun(set(word(A)),$o)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B3: word(A),A7: set(word(A))] :
                  ( aa(set(word(A)),$o,finite_finite2(word(A)),A7)
                 => ( ! [X4: word(A)] :
                        ( aa(set(word(A)),$o,member(word(A),X4),A7)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B3),X4) )
                   => ( aa(set(word(A)),$o,P,A7)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),insert(word(A),B3),A7)) ) ) )
             => aa(set(word(A)),$o,P,A3) ) ) ) ) ).

% signed.finite_linorder_min_induct
tff(fact_7393_signed_Ofinite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),P: fun(set(word(A)),$o)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B3: word(A),A7: set(word(A))] :
                  ( aa(set(word(A)),$o,finite_finite2(word(A)),A7)
                 => ( ! [X4: word(A)] :
                        ( aa(set(word(A)),$o,member(word(A),X4),A7)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X4),B3) )
                   => ( aa(set(word(A)),$o,P,A7)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),insert(word(A),B3),A7)) ) ) )
             => aa(set(word(A)),$o,P,A3) ) ) ) ) ).

% signed.finite_linorder_max_induct
tff(fact_7394_signed_Oinfinite__growing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X5: set(word(A))] :
          ( ( X5 != bot_bot(set(word(A))) )
         => ( ! [X: word(A)] :
                ( aa(set(word(A)),$o,member(word(A),X),X5)
               => ? [Xa2: word(A)] :
                    ( aa(set(word(A)),$o,member(word(A),Xa2),X5)
                    & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Xa2) ) )
           => ~ aa(set(word(A)),$o,finite_finite2(word(A)),X5) ) ) ) ).

% signed.infinite_growing
tff(fact_7395_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,M: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F,N)),aa(nat,word(A),F,aa(nat,nat,suc,N)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F,Nb)),aa(nat,word(A),F,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_7396_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F,N)),aa(nat,word(A),F,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F,Nb)),aa(nat,word(A),F,N3)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_7397_word__sless__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2)) ) ) ).

% word_sless_alt
tff(fact_7398_signed_Ostrict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => sorted_wrt(word(A),word_sless(A),nil(word(A))) ) ).

% signed.strict_sorted_simps(1)
tff(fact_7399_signed_OneqE,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa != Y )
         => ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ) ).

% signed.neqE
tff(fact_7400_signed_Oneq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa != Y )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ) ).

% signed.neq_iff
tff(fact_7401_signed_Oless__asym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ).

% signed.less_asym
tff(fact_7402_signed_Oless__asym_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2) ) ) ).

% signed.less_asym'
tff(fact_7403_signed_Oless__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Z) ) ) ) ).

% signed.less_trans
tff(fact_7404_signed_Oorder_Oasym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2) ) ) ).

% signed.order.asym
tff(fact_7405_signed_Oless__irrefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Xa) ) ).

% signed.less_irrefl
tff(fact_7406_signed_Oless__linear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
          | ( Xa = Y )
          | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ).

% signed.less_linear
tff(fact_7407_signed_Oless__imp__neq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% signed.less_imp_neq
tff(fact_7408_signed_Oless__not__sym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ).

% signed.less_not_sym
tff(fact_7409_signed_Oantisym__conv3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa)
         => ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% signed.antisym_conv3
tff(fact_7410_signed_Oless__imp__triv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),P: $o] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa)
           => (P) ) ) ) ).

% signed.less_imp_triv
tff(fact_7411_signed_Olinorder__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( ( Xa != Y )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ) ).

% signed.linorder_cases
tff(fact_7412_signed_Odual__order_Oasym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2) ) ) ).

% signed.dual_order.asym
tff(fact_7413_signed_Oless__imp__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% signed.less_imp_not_eq
tff(fact_7414_signed_Oless__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ( Y != Xa ) ) ) ).

% signed.less_imp_not_eq2
tff(fact_7415_signed_Odual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),A2) ) ).

% signed.dual_order.irrefl
tff(fact_7416_signed_Oless__imp__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa) ) ) ).

% signed.less_imp_not_less
tff(fact_7417_signed_Oord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( ( A2 = B2 )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),C2) ) ) ) ).

% signed.ord_eq_less_trans
tff(fact_7418_signed_Oord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),C2) ) ) ) ).

% signed.ord_less_eq_trans
tff(fact_7419_signed_Olinorder__less__wlog,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),fun(word(A),$o)),A2: word(A),B2: word(A)] :
          ( ! [A4: word(A),B3: word(A)] :
              ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A4),B3)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B3) )
         => ( ! [A4: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),A4)
           => ( ! [A4: word(A),B3: word(A)] :
                  ( aa(word(A),$o,aa(word(A),fun(word(A),$o),P,B3),A4)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B3) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A2),B2) ) ) ) ) ).

% signed.linorder_less_wlog
tff(fact_7420_signed_Oorder_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),C2) ) ) ) ).

% signed.order.strict_trans
tff(fact_7421_signed_Onot__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Xa)
            | ( Xa = Y ) ) ) ) ).

% signed.not_less_iff_gr_or_eq
tff(fact_7422_signed_Odual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),A2) ) ) ) ).

% signed.dual_order.strict_trans
tff(fact_7423_signed_Oorder_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
         => ( A2 != B2 ) ) ) ).

% signed.order.strict_implies_not_eq
tff(fact_7424_signed_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B2),A2)
         => ( A2 != B2 ) ) ) ).

% signed.dual_order.strict_implies_not_eq
tff(fact_7425_signed_Osorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( distinct(word(A),Xs)
         => distinct(word(A),Xs) ) ) ).

% signed.sorted_list_of_set.distinct_if_distinct_map
tff(fact_7426_signed_Ostrict__sorted__equal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),Ys: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),Xs)
         => ( sorted_wrt(word(A),word_sless(A),Ys)
           => ( ( aa(list(word(A)),set(word(A)),set2(word(A)),Ys) = aa(list(word(A)),set(word(A)),set2(word(A)),Xs) )
             => ( Ys = Xs ) ) ) ) ) ).

% signed.strict_sorted_equal
tff(fact_7427_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,Xa)),aa(int,int,minus_minus(int,ring_1_signed(A,int,Y)),one_one(int))) ) ) ).

% word_sless_sint_le
tff(fact_7428_signed_Ofilter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xs: list(B),Ta: word(A)] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),rev(word(A)),map(B,word(A),F,Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(word(A),fun(B,$o),aTP_Lamp_zb(fun(B,word(A)),fun(word(A),fun(B,$o)),F),Ta)),Xs) = takeWhile(B,aa(word(A),fun(B,$o),aTP_Lamp_zb(fun(B,word(A)),fun(word(A),fun(B,$o)),F),Ta),Xs) ) ) ) ).

% signed.filter_equals_takeWhile_sorted_rev
tff(fact_7429_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),P) = aa(list($o),word(A),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),map(nat,$o,P,upt(zero_zero(nat),type_len0_len_of(A,type2(A))))) ) ).

% word_set_bits_def
tff(fact_7430_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
                & bit_se6407376104438227557le_bit(A,type2(A),Nb) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),M))) ) ).

% drop_bit_exp_eq
tff(fact_7431_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_xs(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_7432_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_xs(nat,$o)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_7433_possible__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( bit_se6407376104438227557le_bit(word(A),type2(word(A)),M)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ).

% possible_bit_word
tff(fact_7434_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,uminus_uminus(A,aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% bit_minus_2_iff
tff(fact_7435_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_7436_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_7437_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,aa(nat,fun(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_7438_word__of__int__conv__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int] : aa(int,word(A),ring_1_of_int(word(A)),I) = bit_bi4170147762399595738t_bits(word(A),bit_se5641148757651400278ts_bit(int,I)) ) ).

% word_of_int_conv_set_bits
tff(fact_7439_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M) ) ) ) ).

% bit_mask_iff
tff(fact_7440_bit__unsigned__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_semiring_bits(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(word(B),A,semiring_1_unsigned(B,A),W)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(word(B),W),Nb) ) ) ) ).

% bit_unsigned_iff
tff(fact_7441_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_7442_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,$o,aa(nat,fun(nat,$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_7443_set__bits__conv__set__bits__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),F) = code_T2661198915054445665ts_aux(A,F,type_len0_len_of(A,type2(A)),zero_zero(word(A))) ) ).

% set_bits_conv_set_bits_aux
tff(fact_7444_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),bit_bi4170147762399595738t_bits(word(A),F)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,F,Nb) ) ) ) ).

% word_test_bit_set_bits
tff(fact_7445_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_7446_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,uminus_uminus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7447_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7448_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_7449_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),aa(nat,nat,aa(nat,fun(nat,nat),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_7450_UNIV__word__eq__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_word_eq_word_of_nat
tff(fact_7451_uint__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),word_rotr(A,Nb),W)) = bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),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_7452_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_xs(nat,$o)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_7453_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),top_top(set(option(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_7454_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      aa(fun(A,$o),set(A),collect(A),aTP_Lamp_zc($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_7455_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_eh(fun(A,$o),fun(A,$o),P)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_7456_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_xr(nat,$o)) = uminus_uminus(int,one_one(int)) ).

% int_set_bits_K_True
tff(fact_7457_boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( uminus_uminus(A,top_top(A)) = bot_bot(A) ) ) ).

% boolean_algebra.compl_one
tff(fact_7458_boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( uminus_uminus(A,bot_bot(A)) = top_top(A) ) ) ).

% boolean_algebra.compl_zero
tff(fact_7459_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = top_top(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),top_top(A))
             => ? [Xa3: A] :
                  ( aa(set(A),$o,member(A,Xa3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa3) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_7460_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_7461_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_eg(A,fun(A,A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_7462_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F),A3)) = top_top(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),top_top(A))
             => ? [Xa3: B] :
                  ( aa(set(B),$o,member(B,Xa3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),aa(B,A,F,Xa3)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_7463_range__constant,axiom,
    ! [B: $tType,A: $tType,Xa: A] : aa(set(B),set(A),image(B,A,aTP_Lamp_bx(A,fun(B,A),Xa)),top_top(set(B))) = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ).

% range_constant
tff(fact_7464_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A2: A,Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Xa) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Xa) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Y) = top_top(A) )
               => ( Xa = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_7465_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_7466_top__option__def,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ( top_top(option(A)) = aa(A,option(A),some(A),top_top(A)) ) ) ).

% top_option_def
tff(fact_7467_word__rotate_Oword__rot__logs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A),Y: word(A)] : aa(word(A),word(A),word_rotr(A,Nb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),word_rotr(A,Nb),Xa)),aa(word(A),word(A),word_rotr(A,Nb),Y)) ) ).

% word_rotate.word_rot_logs(6)
tff(fact_7468_word__rotate_Oword__rot__logs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A),Y: word(A)] : aa(word(A),word(A),word_rotr(A,Nb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),word_rotr(A,Nb),Xa)),aa(word(A),word(A),word_rotr(A,Nb),Y)) ) ).

% word_rotate.word_rot_logs(8)
tff(fact_7469_Compl__UNIV__eq,axiom,
    ! [A: $tType] : uminus_uminus(set(A),top_top(set(A))) = bot_bot(set(A)) ).

% Compl_UNIV_eq
tff(fact_7470_Compl__empty__eq,axiom,
    ! [A: $tType] : uminus_uminus(set(A),bot_bot(set(A))) = top_top(set(A)) ).

% Compl_empty_eq
tff(fact_7471_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xa: A] : top_top(set(A)) != aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_7472_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_7473_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [Xa: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,Xa,Y) = top_top(set(A)) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_7474_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),top_top(A)) ) ).

% top_greatest
tff(fact_7475_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_7476_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_7477_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),top_top(set(A))) ).

% subset_UNIV
tff(fact_7478_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H)) ) ).

% not_UNIV_le_Icc
tff(fact_7479_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A2) ) ).

% top.extremum_strict
tff(fact_7480_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_7481_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H)) ) ).

% not_UNIV_le_Iic
tff(fact_7482_word__rotate_Oword__rot__logs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A),Y: word(A)] : aa(word(A),word(A),word_rotr(A,Nb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),word_rotr(A,Nb),Xa)),aa(word(A),word(A),word_rotr(A,Nb),Y)) ) ).

% word_rotate.word_rot_logs(4)
tff(fact_7483_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_wz(A,$o)) ).

% UNIV_def
tff(fact_7484_word__rotate_Oword__rot__logs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Va2: word(A)] : aa(word(A),word(A),word_rotr(A,Nb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Va2)) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),word_rotr(A,Nb),Va2)) ) ).

% word_rotate.word_rot_logs(2)
tff(fact_7485_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),uminus_uminus(A,Xa)),Y) ) ) ).

% sup_shunt
tff(fact_7486_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_7487_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),B4: set(A),I: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F),top_top(set(B)))),B4)
     => aa(set(A),$o,member(A,aa(B,A,F,I)),B4) ) ).

% range_subsetD
tff(fact_7488_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_zd(fun(C,A),fun(fun(B,C),fun(B,A)),F),G)),top_top(set(B))) = aa(set(C),set(A),image(C,A,F),aa(set(B),set(C),image(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_7489_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F),top_top(set(B))))
     => ~ ! [X: B] : B2 != aa(B,A,F,X) ) ).

% rangeE
tff(fact_7490_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F: fun(A,C)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,G),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_ze(fun(B,A),fun(fun(A,C),fun(B,C)),G),F)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_7491_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A2: A,Xa: B] :
      ( ( aa(set(B),set(A),image(B,A,F),top_top(set(B))) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) )
     => ( aa(B,A,F,Xa) = A2 ) ) ).

% range_eq_singletonD
tff(fact_7492_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),uminus_uminus(set(A),aa(set(B),set(A),image(B,A,F),A3))),aa(set(B),set(A),image(B,A,F),uminus_uminus(set(B),A3))) ) ).

% surj_Compl_image_subset
tff(fact_7493_finite__range__Some,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_range_Some
tff(fact_7494_notin__range__Some,axiom,
    ! [A: $tType,Xa: option(A)] :
      ( ~ aa(set(option(A)),$o,member(option(A),Xa),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A))))
    <=> ( Xa = none(A) ) ) ).

% notin_range_Some
tff(fact_7495_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),A2: B,B2: A] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image(B,option(A),F),top_top(set(B))))
     => aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image(B,option(A),fun_upd(B,option(A),F,A2,aa(A,option(A),some(A),B2))),top_top(set(B)))) ) ).

% finite_range_updI
tff(fact_7496_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = top_top(A) )
           => ( uminus_uminus(A,Xa) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7497_inf__top_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semila1105856199041335345_order(A,inf_inf(A),top_top(A),ord_less_eq(A),ord_less(A)) ) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_7498_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,$o,aa(nat,fun(nat,$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_7499_range__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(set(word(A)),set(int),image(word(A),int,semiring_1_unsigned(A,int)),top_top(set(word(A)))) = set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% range_uint
tff(fact_7500_UNIV__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(int),set(word(A)),image(int,word(A),ring_1_of_int(word(A))),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_eq
tff(fact_7501_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
       => ( aa(set(word(A)),set(word(B)),image(word(A),word(B),semiring_1_unsigned(A,word(B))),top_top(set(word(A)))) = aa(fun(word(B),$o),set(word(B)),collect(word(B)),aTP_Lamp_zf(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_7502_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,W: word(A)] :
          word_roti(A,I,W) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(word(A),word(A),word_rotr(A,nat2(I)),W),aa(word(A),word(A),word_rotl(A,nat2(uminus_uminus(int,I))),W)) ) ).

% word_roti_eq_word_rotr_word_rotl
tff(fact_7503_word__roti_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : word_roti(A,Xa,aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),nat2(modulo_modulo(int,Xa,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),bit_se4197421643247451524op_bit(int,nat2(modulo_modulo(int,Xa,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A))))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)),aa(int,int,bit_se2584673776208193580ke_bit(int,nat2(modulo_modulo(int,Xa,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),Xb))) ) ).

% word_roti.abs_eq
tff(fact_7504_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_7505_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_7506_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_7507_range__mult,axiom,
    ! [A2: real] :
      aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = $ite(A2 = zero_zero(real),aa(set(real),set(real),insert(real,zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_7508_Word__eq__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word2(A) = ring_1_of_int(word(A)) ) ) ).

% Word_eq_word_of_int
tff(fact_7509_word__rot__lr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Va2: word(A)] : aa(word(A),word(A),word_rotr(A,K),aa(word(A),word(A),word_rotl(A,K),Va2)) = Va2 ) ).

% word_rot_lr
tff(fact_7510_word__rot__rl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Va2: word(A)] : aa(word(A),word(A),word_rotl(A,K),aa(word(A),word(A),word_rotr(A,K),Va2)) = Va2 ) ).

% word_rot_rl
tff(fact_7511_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_zg($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_7512_sums__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] : aa(A,$o,sums(A,F),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_zh(fun(nat,A),fun(nat,A),F)),top_top(set(nat))))) ) ).

% sums_SUP
tff(fact_7513_norm__assertion__simps_I4_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),top_top(assn)) = Xa ).

% norm_assertion_simps(4)
tff(fact_7514_norm__assertion__simps_I3_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),top_top(assn)),Xa) = Xa ).

% norm_assertion_simps(3)
tff(fact_7515_ent__star__mono__true,axiom,
    ! [A3: assn,A9: assn,B4: assn,B12: assn] :
      ( entails(A3,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A9),top_top(assn)))
     => ( entails(B4,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B12),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),A9),B12)),top_top(assn))) ) ) ).

% ent_star_mono_true
tff(fact_7516_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_7517_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_7518_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_7519_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_7520_ent__true,axiom,
    ! [P: assn] : entails(P,top_top(assn)) ).

% ent_true
tff(fact_7521_word__rot__gal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Va2: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),word_rotr(A,Nb),Va2) = W )
        <=> ( aa(word(A),word(A),word_rotl(A,Nb),W) = Va2 ) ) ) ).

% word_rot_gal
tff(fact_7522_word__rot__gal_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat,Va2: word(A)] :
          ( ( W = aa(word(A),word(A),word_rotr(A,Nb),Va2) )
        <=> ( Va2 = aa(word(A),word(A),word_rotl(A,Nb),W) ) ) ) ).

% word_rot_gal'
tff(fact_7523_norm__assertion__simps_I12_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),top_top(assn)) = top_top(assn) ).

% norm_assertion_simps(12)
tff(fact_7524_norm__assertion__simps_I11_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),top_top(assn)),Xa) = top_top(assn) ).

% norm_assertion_simps(11)
tff(fact_7525_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_7526_word_Oabs__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Y: int] :
          ( ( aa(int,word(A),word2(A),Xa) = aa(int,word(A),word2(A),Y) )
        <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Y) ) ) ) ).

% word.abs_eq_iff
tff(fact_7527_size__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : aa(word(A),nat,size_size(word(A)),aa(int,word(A),word2(A),Xa)) = type_len0_len_of(A,type2(A)) ) ).

% size_word.abs_eq
tff(fact_7528_UNIV__bool,axiom,
    top_top(set($o)) = aa(set($o),set($o),insert($o,$false),aa(set($o),set($o),insert($o,$true),bot_bot(set($o)))) ).

% UNIV_bool
tff(fact_7529_Un__eq__UN,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_zi(set(A),fun(set(A),fun($o,set(A))),A3),B4)),top_top(set($o)))) ).

% Un_eq_UN
tff(fact_7530_UN__bool__eq,axiom,
    ! [A: $tType,A3: fun($o,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image($o,set(A),A3),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa($o,set(A),A3,$true)),aa($o,set(A),A3,$false)) ).

% UN_bool_eq
tff(fact_7531_SUP__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: fun($o,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set($o),set(A),image($o,A,A3),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa($o,A,A3,$true)),aa($o,A,A3,$false)) ) ).

% SUP_UNIV_bool_expand
tff(fact_7532_surj__list__encode,axiom,
    aa(set(list(nat)),set(nat),image(list(nat),nat,nat_list_encode),top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_7533_surj__prod__encode,axiom,
    aa(set(product_prod(nat,nat)),set(nat),image(product_prod(nat,nat),nat,nat_prod_encode),top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).

% surj_prod_encode
tff(fact_7534_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_7535_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_7536_push__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : bit_se4730199178511100633sh_bit(word(A),Xa,aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_se4730199178511100633sh_bit(int,Xa,Xb)) ) ).

% push_bit_word.abs_eq
tff(fact_7537_minus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),minus_minus(word(A),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,minus_minus(int,Xa),Xb)) ) ).

% minus_word.abs_eq
tff(fact_7538_word__rotate_Oword__rot__logs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Va2: word(A)] : aa(word(A),word(A),word_rotl(A,Nb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Va2)) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),word_rotl(A,Nb),Va2)) ) ).

% word_rotate.word_rot_logs(1)
tff(fact_7539_not__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(int,word(A),word2(A),Xa)) = aa(int,word(A),word2(A),aa(int,int,bit_ri4277139882892585799ns_not(int),Xa)) ) ).

% not_word.abs_eq
tff(fact_7540_one__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = aa(int,word(A),word2(A),one_one(int)) ) ) ).

% one_word_def
tff(fact_7541_zero__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = aa(int,word(A),word2(A),zero_zero(int)) ) ) ).

% zero_word_def
tff(fact_7542_uminus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : uminus_uminus(word(A),aa(int,word(A),word2(A),Xa)) = aa(int,word(A),word2(A),uminus_uminus(int,Xa)) ) ).

% uminus_word.abs_eq
tff(fact_7543_mask__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] : bit_se2239418461657761734s_mask(word(A),Xa) = aa(int,word(A),word2(A),bit_se2239418461657761734s_mask(int,Xa)) ) ).

% mask_word.abs_eq
tff(fact_7544_set__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : bit_se5668285175392031749et_bit(word(A),Xa,aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_se5668285175392031749et_bit(int,Xa,Xb)) ) ).

% set_bit_word.abs_eq
tff(fact_7545_word_Oabs__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),Xa: word(A)] :
          ( ! [Y3: int] : aa(word(A),$o,P,aa(int,word(A),word2(A),Y3))
         => aa(word(A),$o,P,Xa) ) ) ).

% word.abs_induct
tff(fact_7546_unset__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : bit_se2638667681897837118et_bit(word(A),Xa,aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_se2638667681897837118et_bit(int,Xa,Xb)) ) ).

% unset_bit_word.abs_eq
tff(fact_7547_flip__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : bit_se8732182000553998342ip_bit(word(A),Xa,aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_se8732182000553998342ip_bit(int,Xa,Xb)) ) ).

% flip_bit_word.abs_eq
tff(fact_7548_word__rotate_Oword__rot__logs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A),Y: word(A)] : aa(word(A),word(A),word_rotl(A,Nb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),word_rotl(A,Nb),Xa)),aa(word(A),word(A),word_rotl(A,Nb),Y)) ) ).

% word_rotate.word_rot_logs(3)
tff(fact_7549_and__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Xb)) ) ).

% and_word.abs_eq
tff(fact_7550_word__rotate_Oword__rot__logs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A),Y: word(A)] : aa(word(A),word(A),word_rotl(A,Nb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),word_rotl(A,Nb),Xa)),aa(word(A),word(A),word_rotl(A,Nb),Y)) ) ).

% word_rotate.word_rot_logs(7)
tff(fact_7551_xor__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Xb)) ) ).

% xor_word.abs_eq
tff(fact_7552_word__rotate_Oword__rot__logs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xa: word(A),Y: word(A)] : aa(word(A),word(A),word_rotl(A,Nb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),word_rotl(A,Nb),Xa)),aa(word(A),word(A),word_rotl(A,Nb),Y)) ) ).

% word_rotate.word_rot_logs(5)
tff(fact_7553_or__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Xb)) ) ).

% or_word.abs_eq
tff(fact_7554_times__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),times_times(int),Xa),Xb)) ) ).

% times_word.abs_eq
tff(fact_7555_plus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Xb)) ) ).

% plus_word.abs_eq
tff(fact_7556_word__succ_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : word_succ(A,aa(int,word(A),word2(A),Xa)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int))) ) ).

% word_succ.abs_eq
tff(fact_7557_word__pred_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : word_pred(A,aa(int,word(A),word2(A),Xa)) = aa(int,word(A),word2(A),aa(int,int,minus_minus(int,Xa),one_one(int))) ) ).

% word_pred.abs_eq
tff(fact_7558_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_7559_word__rotr__rev,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] : aa(word(A),word(A),word_rotr(A,Nb),W) = word_reverse(A,aa(word(A),word(A),word_rotl(A,Nb),word_reverse(A,W))) ) ).

% word_rotr_rev
tff(fact_7560_word__rotl_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : aa(word(A),word(A),word_rotl(A,Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_concat_bit(modulo_modulo(nat,Xa,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,Xa,type_len0_len_of(A,type2(A)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),modulo_modulo(nat,Xa,type_len0_len_of(A,type2(A))))),Xb))) ) ).

% word_rotl.abs_eq
tff(fact_7561_word__int__case_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Xa: fun(int,A),Xb: int] : word_int_case(A,B,Xa,aa(int,word(B),word2(B),Xb)) = aa(int,A,Xa,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),Xb)) ) ).

% word_int_case.abs_eq
tff(fact_7562_less__eq__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)) ) ) ).

% less_eq_word.abs_eq
tff(fact_7563_less__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)) ) ) ).

% less_word.abs_eq
tff(fact_7564_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_7565_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,X4: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(word(A),aa(int,word(A),word2(A),Xa)),X4)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,Xa),X4) ) ) ) ).

% bit_word.abs_eq
tff(fact_7566_divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb))) ) ).

% divide_word.abs_eq
tff(fact_7567_modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : modulo_modulo(word(A),aa(int,word(A),word2(A),Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),modulo_modulo(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb))) ) ).

% modulo_word.abs_eq
tff(fact_7568_take__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),Xa)),Xb)) ) ).

% take_bit_word.abs_eq
tff(fact_7569_drop__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : bit_se4197421643247451524op_bit(word(A),Xa,aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_se4197421643247451524op_bit(int,Xa,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb))) ) ).

% drop_bit_word.abs_eq
tff(fact_7570_udvd_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( udvd(A,aa(int,word(A),word2(A),Xa),aa(int,word(A),word2(A),Xb))
        <=> aa(int,$o,dvd_dvd(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)) ) ) ).

% udvd.abs_eq
tff(fact_7571_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aTP_Lamp_zj(fun(nat,set(A)),fun(nat,set(A)),A3)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_7572_word__cat_Oabs__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [Xa: int,Xb: int] : word_cat(B,C,A,aa(int,word(B),word2(B),Xa),aa(int,word(C),word2(C),Xb)) = aa(int,word(A),word2(A),bit_concat_bit(type_len0_len_of(C,type2(C)),Xb,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),Xa))) ) ).

% word_cat.abs_eq
tff(fact_7573_unsigned_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Xa: int] : aa(word(B),A,semiring_1_unsigned(B,A),aa(int,word(B),word2(B),Xa)) = aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),Xa))) ) ).

% unsigned.abs_eq
tff(fact_7574_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C3: set(A)] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),C3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),C3) ) ).

% UN_finite_subset
tff(fact_7575_UN__finite2__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_7576_suminf__eq__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] : suminf(A,F) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_zh(fun(nat,A),fun(nat,A),F)),top_top(set(nat)))) ) ).

% suminf_eq_SUP
tff(fact_7577_range__mod,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_zk(nat,fun(nat,nat),Nb)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_7578_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_7579_UN__finite2__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_7580_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb)) ) ) ).

% word_sle.abs_eq
tff(fact_7581_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb))
        <=> aa(int,$o,aa(int,fun(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)))),Xa)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb)) ) ) ).

% word_sless.abs_eq
tff(fact_7582_signed_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Xa: int] : ring_1_signed(B,A,aa(int,word(B),word2(B),Xa)) = 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)))),Xa)) ) ).

% signed.abs_eq
tff(fact_7583_suminf__eq__SUP__real,axiom,
    ! [X5: fun(nat,real)] :
      ( summable(real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,X5,I2))
       => ( suminf(real,X5) = aa(set(real),real,complete_Sup_Sup(real),aa(set(nat),set(real),image(nat,real,aTP_Lamp_zl(fun(nat,real),fun(nat,real),X5)),top_top(set(nat)))) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_7584_signed__drop__bit_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : signed_drop_bit(A,Xa,aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_se4197421643247451524op_bit(int,Xa,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb))) ) ).

% signed_drop_bit.abs_eq
tff(fact_7585_signed__divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : signed7115095781618012415divide(word(A),aa(int,word(A),word2(A),Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),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)))),Xa),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb))) ) ).

% signed_divide_word.abs_eq
tff(fact_7586_word__rotr_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : aa(word(A),word(A),word_rotr(A,Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),modulo_modulo(nat,Xa,type_len0_len_of(A,type2(A)))),bit_se4197421643247451524op_bit(int,modulo_modulo(nat,Xa,type_len0_len_of(A,type2(A))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)),aa(int,int,bit_se2584673776208193580ke_bit(int,modulo_modulo(nat,Xa,type_len0_len_of(A,type2(A)))),Xb))) ) ).

% word_rotr.abs_eq
tff(fact_7587_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,$o,aa(nat,fun(nat,$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_7588_signed__cast_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: int] : signed_cast(B,A,aa(int,word(B),word2(B),Xa)) = aa(int,word(A),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)))),Xa)) ) ).

% signed_cast.abs_eq
tff(fact_7589_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : the_signed_int(A,aa(int,word(A),word2(A),Xa)) = 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)))),Xa) ) ).

% the_signed_int.abs_eq
tff(fact_7590_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_7591_of__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] : of_nat(A,Xa) = aa(int,word(A),word2(A),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),Xa))) ) ).

% of_nat.abs_eq
tff(fact_7592_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_zm(assn,fun(A,assn),C2)) = C2 ).

% ex_assn_const
tff(fact_7593_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_zn(assn,fun(fun(A,assn),fun(A,assn)),R),Q)) ).

% norm_assertion_simps(17)
tff(fact_7594_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_aa(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ).

% norm_assertion_simps(16)
tff(fact_7595_triv__exI,axiom,
    ! [A: $tType,Q: fun(A,assn),Xa: A] : entails(aa(A,assn,Q,Xa),ex_assn(A,Q)) ).

% triv_exI
tff(fact_7596_norm__assertion__simps_I20_J,axiom,
    ! [A: $tType,Q: fun(A,assn),P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,Q)),P) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zo(fun(A,assn),fun(assn,fun(A,assn)),Q),P)) ).

% norm_assertion_simps(20)
tff(fact_7597_norm__assertion__simps_I21_J,axiom,
    ! [A: $tType,Q: assn,P: fun(A,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Q),ex_assn(A,P)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zp(assn,fun(fun(A,assn),fun(A,assn)),Q),P)) ).

% norm_assertion_simps(21)
tff(fact_7598_norm__assertion__simps_I19_J,axiom,
    ! [A: $tType,Q: assn,P: fun(A,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Q),ex_assn(A,P)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zq(assn,fun(fun(A,assn),fun(A,assn)),Q),P)) ).

% norm_assertion_simps(19)
tff(fact_7599_norm__assertion__simps_I18_J,axiom,
    ! [A: $tType,Q: fun(A,assn),P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,Q)),P) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zr(fun(A,assn),fun(assn,fun(A,assn)),Q),P)) ).

% norm_assertion_simps(18)
tff(fact_7600_norm__post__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: assn,F: heap_Time_Heap(A),Q: fun(B,fun(A,assn)),Xa: B,Ta: nat] :
      ( time_htt(A,P,F,aa(B,fun(A,assn),Q,Xa),Ta)
     => time_htt(A,P,F,aTP_Lamp_zt(fun(B,fun(A,assn)),fun(A,assn),Q),Ta) ) ).

% norm_post_ex_rule_htt
tff(fact_7601_norm__pre__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: fun(A,assn),F: heap_Time_Heap(B),Q: fun(B,assn),Ta: nat] :
      ( ! [X: A] : time_htt(B,aa(A,assn,P,X),F,Q,Ta)
     => time_htt(B,ex_assn(A,P),F,Q,Ta) ) ).

% norm_pre_ex_rule_htt
tff(fact_7602_enorm__exI_H,axiom,
    ! [A: $tType,Z5: fun(A,$o),P: assn,Q: fun(A,assn)] :
      ( ! [X: A] :
          ( aa(A,$o,Z5,X)
         => entails(P,aa(A,assn,Q,X)) )
     => ( ? [X_1: A] : aa(A,$o,Z5,X_1)
       => entails(P,ex_assn(A,Q)) ) ) ).

% enorm_exI'
tff(fact_7603_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_7604_ent__ex__postI,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xa: A] :
      ( entails(P,aa(A,assn,Q,Xa))
     => entails(P,ex_assn(A,Q)) ) ).

% ent_ex_postI
tff(fact_7605_ex__join__or,axiom,
    ! [A: $tType,P: fun(A,assn),Q: fun(A,assn)] : ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zu(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zv(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) ).

% ex_join_or
tff(fact_7606_ex__distrib__or,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zo(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,P)),Q) ).

% ex_distrib_or
tff(fact_7607_post__exI__rule,axiom,
    ! [B: $tType,A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,fun(B,assn)),Xa: B] :
      ( hoare_hoare_triple(A,P,C2,aa(B,fun(A,assn),aTP_Lamp_zw(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Q),Xa))
     => hoare_hoare_triple(A,P,C2,aTP_Lamp_zx(fun(A,fun(B,assn)),fun(A,assn),Q)) ) ).

% post_exI_rule
tff(fact_7608_norm__pre__ex__rule,axiom,
    ! [A: $tType,B: $tType,P: fun(A,assn),F: heap_Time_Heap(B),Q: fun(B,assn)] :
      ( ! [X: A] : hoare_hoare_triple(B,aa(A,assn,P,X),F,Q)
     => hoare_hoare_triple(B,ex_assn(A,P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_7609_ex__distrib__star,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_aa(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_7610_ex__distrib__and,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_zr(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,P)),Q) ).

% ex_distrib_and
tff(fact_7611_the__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : the_nat(A,aa(int,word(A),word2(A),Xa)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa)) ) ).

% the_nat.abs_eq
tff(fact_7612_cast_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: int] : cast(B,A,aa(int,word(B),word2(B),Xa)) = aa(int,word(A),word2(A),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),Xa)) ) ).

% cast.abs_eq
tff(fact_7613_Word_Oof__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : of_int(A,Xa) = aa(int,word(A),word2(A),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa)) ) ).

% Word.of_int.abs_eq
tff(fact_7614_bin__last__set__bits,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,F))
      <=> aa(nat,$o,F,zero_zero(nat)) ) ) ).

% bin_last_set_bits
tff(fact_7615_wf__set__bits__int__Suc,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_zy(fun(nat,$o),fun(nat,$o),F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_7616_ones,axiom,
    ! [Nb: nat,F: fun(nat,$o)] :
      ( ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
         => aa(nat,$o,F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% ones
tff(fact_7617_wf__set__bits__int_Ocases,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( ! [N: nat] :
            ~ ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N4)
               => ~ aa(nat,$o,F,N4) )
       => ~ ! [N: nat] :
              ~ ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N4)
                 => aa(nat,$o,F,N4) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_7618_wf__set__bits__int_Osimps,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
    <=> ( ? [N6: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
           => ~ aa(nat,$o,F,N11) )
        | ? [N6: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
           => aa(nat,$o,F,N11) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_7619_zeros,axiom,
    ! [Nb: nat,F: fun(nat,$o)] :
      ( ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
         => ~ aa(nat,$o,F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% zeros
tff(fact_7620_wf__set__bits__int__simps,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
    <=> ? [N6: nat] :
          ( ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
             => ~ aa(nat,$o,F,N11) )
          | ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
             => aa(nat,$o,F,N11) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_7621_wf__set__bits__int__const,axiom,
    ! [B2: $o] : bit_wf_set_bits_int(aTP_Lamp_zz($o,fun(nat,$o),(B2))) ).

% wf_set_bits_int_const
tff(fact_7622_the__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : the_int(A,aa(int,word(A),word2(A),Xa)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa) ) ).

% the_int.abs_eq
tff(fact_7623_int__set__bits__unfold__BIT,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi4170147762399595738t_bits(int,F) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,F,zero_zero(nat)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_7624_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),map(B,A,F,Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),filter2(B,comp(A,$o,B,P,F)),Xs)) ).

% length_filter_map
tff(fact_7625_bin__rest__set__bits,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_bi4170147762399595738t_bits(int,F)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_7626_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,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(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_7627_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,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(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_7628_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_7629_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_7630_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,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X)),aa(B,A,H,Y3))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),comp(B,A,C,H,G)),A3) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_7631_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),Q: fun(B,$o),G: fun(A,B),X4: A] :
      ( aa(A,$o,comp(B,$o,A,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aaa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),P),Q),G),X4)
    <=> ( aa(A,$o,comp(B,$o,A,P,G),X4)
        & aa(A,$o,comp(B,$o,A,Q,G),X4) ) ) ).

% conj_comp_iff
tff(fact_7632_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,comp(A,A,nat,inverse_inverse(A),F))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_aab(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_inverse_divide
tff(fact_7633_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,Y3: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => ( aa(set(A),$o,member(A,Y3),A3)
                 => ( ( X != Y3 )
                   => ( ( aa(A,B,H,X) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(A),set(B),image(A,B,H),A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),comp(B,C,A,G,H)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_7634_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F,I2))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),aa(set(A),set(C),image(A,C,F),I5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),comp(C,B,A,G,F)),I5)) ) ) ) ).

% sum_image_le
tff(fact_7635_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(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_7636_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(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_7637_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_7638_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_7639_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(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_7640_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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_7641_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,aTP_Lamp_le(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_7642_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),comp(nat,A,nat,G,aTP_Lamp_le(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_7643_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,aTP_Lamp_le(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_7644_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,aTP_Lamp_le(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_7645_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(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_7646_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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_7647_signed__modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : signed6721504322012087516modulo(word(A),aa(int,word(A),word2(A),Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),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)))),Xa),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb))) ) ).

% signed_modulo_word.abs_eq
tff(fact_7648_signed_Oinsort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [A2: A,Xs: list(A),F: fun(A,word(B))] :
          ( aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(word(B),word_sle(B),map(A,word(B),F,Xs))
           => ( ( aa(list(A),A,hd(A),aa(list(A),list(A),filter2(A,aa(fun(A,word(B)),fun(A,$o),aTP_Lamp_aac(A,fun(fun(A,word(B)),fun(A,$o)),A2),F)),Xs)) = A2 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ) ).

% signed.insort_key_remove1
tff(fact_7649_smod__word__mod__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : signed6721504322012087516modulo(word(A),Xa,zero_zero(word(A))) = Xa ) ).

% smod_word_mod_0
tff(fact_7650_smod__word__0__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),Xa) = zero_zero(word(A)) ) ).

% smod_word_0_mod
tff(fact_7651_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_7652_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_7653_signed_Oremove1__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Xa: A,F: fun(A,word(B)),Xs: list(A)] : remove1(A,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),Xs)) = Xs ) ).

% signed.remove1_insort_key
tff(fact_7654_signed_Olength__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F: fun(A,word(B)),Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% signed.length_insort
tff(fact_7655_smod__word__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,uminus_uminus(word(A),one_one(word(A)))) = zero_zero(word(A)) ) ).

% smod_word_minus_one
tff(fact_7656_signed_Osorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] : comp(list(word(A)),list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Y),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa)) = comp(list(word(A)),list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Y)) ) ).

% signed.sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_7657_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_7658_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] : comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Y),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa)) = comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Xa),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_dw(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_7659_in__Union__o__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xa: A,Gset: fun(B,set(set(A))),Gmap: fun(C,B),A3: C] :
      ( aa(set(A),$o,member(A,Xa),aa(C,set(A),comp(B,set(A),C,comp(set(set(A)),set(A),B,complete_Sup_Sup(set(A)),Gset),Gmap),A3))
     => aa(set(A),$o,member(A,Xa),aa(C,set(A),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A)),comp(B,set(set(A)),C,Gset,Gmap)),A3)) ) ).

% in_Union_o_assoc
tff(fact_7660_empty__natural,axiom,
    ! [C: $tType,B: $tType,D6: $tType,A: $tType,F: fun(A,C),G: fun(D6,B)] : comp(C,set(B),A,aTP_Lamp_aae(C,set(B)),F) = comp(set(D6),set(B),A,image(D6,B,G),aTP_Lamp_aaf(A,set(D6))) ).

% empty_natural
tff(fact_7661_Union__natural,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : comp(set(set(B)),set(B),set(set(A)),complete_Sup_Sup(set(B)),image(set(A),set(B),image(A,B,F))) = comp(set(A),set(B),set(set(A)),image(A,B,F),complete_Sup_Sup(set(A))) ).

% Union_natural
tff(fact_7662_signed_Oinsort__not__Nil,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F: fun(A,word(B)),A2: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),A2),Xs) != nil(A) ) ).

% signed.insort_not_Nil
tff(fact_7663_signed_Ofilter__insort__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),Xa: A,F: fun(A,word(B)),Xs: list(A)] :
          ( ~ aa(A,$o,P,Xa)
         => ( aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ) ).

% signed.filter_insort_triv
tff(fact_7664_signed_Osorted__insort,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),Xs))
        <=> sorted_wrt(word(A),word_sle(A),Xs) ) ) ).

% signed.sorted_insort
tff(fact_7665_signed_Oinsort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F: fun(A,word(B)),Xa: A,Y: A,Ys: list(A)] :
          aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),aa(list(A),list(A),cons(A,Y),Ys)) = $ite(aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F,Xa)),aa(A,word(B),F,Y)),aa(list(A),list(A),cons(A,Xa),aa(list(A),list(A),cons(A,Y),Ys)),aa(list(A),list(A),cons(A,Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),Ys))) ) ).

% signed.insort_key.simps(2)
tff(fact_7666_signed_Oinsort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xa: B,Y: B,Xs: list(B)] :
          ( ( aa(B,word(A),F,Xa) != aa(B,word(A),F,Y) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Xa),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Xa),aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Y),Xs)) ) ) ) ).

% signed.insort_key_left_comm
tff(fact_7667_signed_Oinsort__left__comm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Xs: list(word(A))] : aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Y),Xs)) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Y),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),Xs)) ) ).

% signed.insort_left_comm
tff(fact_7668_signed_Oinsort__key_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F: fun(A,word(B)),Xa: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),nil(A)) = aa(list(A),list(A),cons(A,Xa),nil(A)) ) ).

% signed.insort_key.simps(1)
tff(fact_7669_signed_Oinsort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Xs: list(A),F: fun(A,word(B)),A2: A] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F,A2)),aa(A,word(B),F,X)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),A2),Xs) = aa(list(A),list(A),cons(A,A2),Xs) ) ) ) ).

% signed.insort_is_Cons
tff(fact_7670_signed_Oset__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F: fun(A,word(B)),Xa: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),Xs)) = aa(set(A),set(A),insert(A,Xa),aa(list(A),set(A),set2(A),Xs)) ) ).

% signed.set_insort_key
tff(fact_7671_sum_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B4: set(set(A)),G: fun(A,B)] :
          ( ! [X: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X),B4)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [A12: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),A12),B4)
               => ! [A23: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),A23),B4)
                   => ( ( A12 != A23 )
                     => ! [X: A] :
                          ( aa(set(A),$o,member(A,X),A12)
                         => ( aa(set(A),$o,member(A,X),A23)
                           => ( aa(A,B,G,X) = zero_zero(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B),groups7311177749621191930dd_sum(A,B)),G),B4) ) ) ) ) ).

% sum.Union_comp
tff(fact_7672_signed_Odistinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F: fun(A,word(B)),Xa: A,Xs: list(A)] :
          ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),Xs))
        <=> ( ~ aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
            & distinct(A,Xs) ) ) ) ).

% signed.distinct_insort
tff(fact_7673_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(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_7674_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),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_7675_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S3: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,G),top_top(set(C)))),S3)
       => finite4664212375090638736ute_on(C,B,R,comp(A,fun(B,B),C,F,G)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_7676_signed_Osorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xa: B,Xs: list(B)] :
          ( sorted_wrt(word(A),word_sle(A),map(B,word(A),F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Xa),Xs)))
        <=> sorted_wrt(word(A),word_sle(A),map(B,word(A),F,Xs)) ) ) ).

% signed.sorted_insort_key
tff(fact_7677_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S3: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S3,F)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,G),top_top(set(C)))),S3)
       => finite673082921795544331dem_on(C,B,R,comp(A,fun(B,B),C,F,G)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_7678_sum_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(set(A)),G: fun(A,B)] :
          ( ! [X: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X),C3)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [X: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),X),C3)
               => ! [Xa4: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),Xa4),C3)
                   => ( ( X != Xa4 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa4) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B),groups7311177749621191930dd_sum(A,B)),G),C3) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_7679_prod_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(set(A)),G: fun(A,B)] :
          ( ! [X: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X),C3)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [X: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),X),C3)
               => ! [Xa4: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),Xa4),C3)
                   => ( ( X != Xa4 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa4) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B),groups7121269368397514597t_prod(A,B)),G),C3) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_7680_signed_Ofilter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xs: list(B),P: fun(B,$o),Xa: B] :
          ( sorted_wrt(word(A),word_sle(A),map(B,word(A),F,Xs))
         => ( aa(B,$o,P,Xa)
           => ( aa(list(B),list(B),filter2(B,P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Xa),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Xa),aa(list(B),list(B),filter2(B,P),Xs)) ) ) ) ) ).

% signed.filter_insort
tff(fact_7681_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_7682_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_7683_SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3)) = finite_fold(A,B,comp(B,fun(B,B),A,sup_sup(B),F),bot_bot(B),A3) ) ) ) ).

% SUP_fold_sup
tff(fact_7684_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F,Xs) = map(B,A,comp(option(A),A,B,the2(A),F),aa(list(B),list(B),filter2(B,aTP_Lamp_aag(fun(B,option(A)),fun(B,$o),F)),Xs)) ).

% map_filter_def
tff(fact_7685_signed_Oinsort__remove1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Xs: list(word(A))] :
          ( aa(set(word(A)),$o,member(word(A),A2),aa(list(word(A)),set(word(A)),set2(word(A)),Xs))
         => ( sorted_wrt(word(A),word_sle(A),Xs)
           => ( aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),A2),remove1(word(A),A2,Xs)) = Xs ) ) ) ) ).

% signed.insort_remove1
tff(fact_7686_word__rcat__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( word_rcat(A,B) = comp(list(word(A)),word(B),list(word(A)),comp(int,word(B),list(word(A)),ring_1_of_int(word(B)),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))))),rev(word(A))) ) ) ).

% word_rcat_def
tff(fact_7687_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_7688_word__smod__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Xa),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)),Xa)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_smod_numerals(4)
tff(fact_7689_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_7690_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_7691_signed_Odistinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xa: B,Xs: list(B)] :
          ( distinct(word(A),map(B,word(A),F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Xa),Xs)))
        <=> ( ~ aa(set(word(A)),$o,member(word(A),aa(B,word(A),F,Xa)),aa(set(B),set(word(A)),image(B,word(A),F),aa(list(B),set(B),set2(B),Xs)))
            & distinct(word(A),map(B,word(A),F,Xs)) ) ) ) ).

% signed.distinct_insort_key
tff(fact_7692_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N6: nat] :
            ( aa(nat,$o,aa(nat,fun(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_7693_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
                 => ( aa(nat,$o,aa(nat,fun(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_7694_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_7695_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_7696_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_7697_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_7698_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N6: nat] :
            ( aa(nat,$o,aa(nat,fun(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_7699_signed_Osorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( sorted_list_of_set(word(A),word_sle(A),aa(set(word(A)),set(word(A)),insert(word(A),Xa),A3)) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),sorted_list_of_set(word(A),word_sle(A),aa(set(word(A)),set(word(A)),minus_minus(set(word(A)),A3),aa(set(word(A)),set(word(A)),insert(word(A),Xa),bot_bot(set(word(A))))))) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_7700_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Nb)
                & aa(nat,$o,aa(nat,fun(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_7701_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_7702_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_7703_signed_Osorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( sorted_list_of_set(word(A),word_sle(A),bot_bot(set(word(A)))) = nil(word(A)) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_7704_signed_Osorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( ~ aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( sorted_list_of_set(word(A),word_sle(A),A3) = nil(word(A)) ) ) ) ).

% signed.sorted_list_of_set.fold_insort_key.infinite
tff(fact_7705_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_7706_signed_Osorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(list(word(A)),set(word(A)),set2(word(A)),sorted_list_of_set(word(A),word_sle(A),A3)) = A3 ) ) ) ).

% signed.sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_7707_sshiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Nb: nat] : bit_Sh8784991116023147202shiftr(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),uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_Suc
tff(fact_7708_signed_Osorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( sorted_list_of_set(word(A),word_sle(A),A3) = nil(word(A)) )
          <=> ( A3 = bot_bot(set(word(A))) ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7709_signed_Osorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ~ aa(set(word(A)),$o,member(word(A),Xa),A3)
           => ( sorted_list_of_set(word(A),word_sle(A),aa(set(word(A)),set(word(A)),insert(word(A),Xa),A3)) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),sorted_list_of_set(word(A),word_sle(A),A3)) ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_7710_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_7711_signed_Osorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] : sorted_wrt(word(A),word_sle(A),sorted_list_of_set(word(A),word_sle(A),A3)) ) ).

% signed.sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_7712_signed_Osorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] : distinct(word(A),sorted_list_of_set(word(A),word_sle(A),A3)) ) ).

% signed.sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_7713_signed_Osorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),B4: set(word(A))] :
          ( ( sorted_list_of_set(word(A),word_sle(A),A3) = sorted_list_of_set(word(A),word_sle(A),B4) )
         => ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),B4)
             => ( A3 = B4 ) ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_7714_signed_Osorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] : sorted_wrt(word(A),word_sless(A),sorted_list_of_set(word(A),word_sle(A),A3)) ) ).

% signed.sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_7715_signed_Osorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] : sorted_list_of_set(word(A),word_sle(A),A3) = finite_fold(word(A),list(word(A)),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),nil(word(A)),A3) ) ).

% signed.sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_7716_signed_Osorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => ( distinct(word(A),Xs)
           => ( sorted_list_of_set(word(A),word_sle(A),aa(list(word(A)),set(word(A)),set2(word(A)),Xs)) = Xs ) ) ) ) ).

% signed.sorted_list_of_set.idem_if_sorted_distinct
tff(fact_7717_signed_Osorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( aa(set(word(A)),$o,member(word(A),Xa),A3)
           => ( sorted_list_of_set(word(A),word_sle(A),A3) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),sorted_list_of_set(word(A),word_sle(A),aa(set(word(A)),set(word(A)),minus_minus(set(word(A)),A3),aa(set(word(A)),set(word(A)),insert(word(A),Xa),bot_bot(set(word(A))))))) ) ) ) ) ).

% signed.sorted_list_of_set.fold_insort_key.remove
tff(fact_7718_signed_Osorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),Xa: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( sorted_list_of_set(word(A),word_sle(A),aa(set(word(A)),set(word(A)),minus_minus(set(word(A)),A3),aa(set(word(A)),set(word(A)),insert(word(A),Xa),bot_bot(set(word(A)))))) = remove1(word(A),Xa,sorted_list_of_set(word(A),word_sle(A),A3)) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_7719_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_7720_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_7721_shiftl__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_Sh4282982442137083160shiftl(A,zero_zero(A),Nb) = zero_zero(A) ) ).

% shiftl_0
tff(fact_7722_shiftl__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083160shiftl(A,A2,zero_zero(nat)) = A2 ) ).

% shiftl_of_0
tff(fact_7723_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_7724_shiftl__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Nb: nat] : bit_Sh4282982442137083160shiftl(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),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_Suc
tff(fact_7725_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xa: A,Nb: nat] : bit_Sh4282982442137083160shiftl(A,Xa,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_7726_bit__shiftl__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_Sh4282982442137083160shiftl(A,A2,M)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
            & bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ) ).

% bit_shiftl_iff
tff(fact_7727_shiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Nb: nat] : bit_Sh4282982442137083166shiftr(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),uminus_uminus(A,aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_Suc
tff(fact_7728_extract__def,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aah(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),dropWhile(A,comp($o,$o,A,fNot,P),Xs)) ).

% extract_def
tff(fact_7729_shiftr__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_Sh4282982442137083166shiftr(A,zero_zero(A),Nb) = zero_zero(A) ) ).

% shiftr_0
tff(fact_7730_shiftr__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083166shiftr(A,A2,zero_zero(nat)) = A2 ) ).

% shiftr_of_0
tff(fact_7731_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_7732_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_7733_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_7734_signed_Osorted__dropWhile,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),P: fun(word(A),$o)] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => sorted_wrt(word(A),word_sle(A),dropWhile(word(A),P,Xs)) ) ) ).

% signed.sorted_dropWhile
tff(fact_7735_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_7736_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_dropWhile_le
tff(fact_7737_dropWhile__eq__drop,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : dropWhile(A,P,Xs) = drop(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% dropWhile_eq_drop
tff(fact_7738_length__dropWhile__takeWhile,axiom,
    ! [A: $tType,Xa: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_dropWhile_takeWhile
tff(fact_7739_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_7740_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
       => ( dropWhile(A,aTP_Lamp_li(A,fun(A,$o),Xa),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),cons(A,Xa),aa(list(A),list(A),rev(A),takeWhile(A,aTP_Lamp_li(A,fun(A,$o),Xa),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_7741_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs))
       => ( takeWhile(A,aTP_Lamp_li(A,fun(A,$o),Xa),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),dropWhile(A,aTP_Lamp_li(A,fun(A,$o),Xa),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_7742_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_aai(A,fun(list(A),option(A)))),dropWhile(A,comp($o,$o,A,fNot,P),Xs)) ).

% find_dropWhile
tff(fact_7743_signed_Oinsort__insert__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xa: B,Xs: list(B)] :
          ( ~ aa(set(word(A)),$o,member(word(A),aa(B,word(A),F,Xa)),aa(set(B),set(word(A)),image(B,word(A),F),aa(list(B),set(B),set2(B),Xs)))
         => ( insort_insert_key(word(A),B,word_sle(A),F,Xa,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),insort_key(word(A),B,word_sle(A),F),Xa),Xs) ) ) ) ).

% signed.insort_insert_insort_key
tff(fact_7744_find__SomeD_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Xa: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),Xa) )
     => aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Xs)) ) ).

% find_SomeD(2)
tff(fact_7745_find__SomeD_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Xa: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),Xa) )
     => aa(A,$o,P,Xa) ) ).

% find_SomeD(1)
tff(fact_7746_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xa: A,Xs: list(A)] :
      find(A,P,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(aa(A,$o,P,Xa),aa(A,option(A),some(A),Xa),find(A,P,Xs)) ).

% find.simps(2)
tff(fact_7747_signed_Osorted__insort__insert,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),Xa: word(A)] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => sorted_wrt(word(A),word_sle(A),insort_insert_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A)),Xa,Xs)) ) ) ).

% signed.sorted_insort_insert
tff(fact_7748_signed_Odistinct__insort__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Xs: list(A),F: fun(A,word(B)),Xa: A] :
          ( distinct(A,Xs)
         => distinct(A,insort_insert_key(word(B),A,word_sle(B),F,Xa,Xs)) ) ) ).

% signed.distinct_insort_insert
tff(fact_7749_signed_Oinsort__insert__triv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Xs: list(word(A))] :
          ( aa(set(word(A)),$o,member(word(A),Xa),aa(list(word(A)),set(word(A)),set2(word(A)),Xs))
         => ( insort_insert_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A)),Xa,Xs) = Xs ) ) ) ).

% signed.insort_insert_triv
tff(fact_7750_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Xa: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),Xa) )
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I3))
          & ( Xa = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_7751_find__Some__iff2,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),Xa) = find(A,P,Xs) )
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I3))
          & ( Xa = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_7752_signed_Oset__insort__insert,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Xs: list(word(A))] : aa(list(word(A)),set(word(A)),set2(word(A)),insort_insert_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A)),Xa,Xs)) = aa(set(word(A)),set(word(A)),insert(word(A),Xa),aa(list(word(A)),set(word(A)),set2(word(A)),Xs)) ) ).

% signed.set_insort_insert
tff(fact_7753_signed_Oinsort__insert__insort,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Xs: list(word(A))] :
          ( ~ aa(set(word(A)),$o,member(word(A),Xa),aa(list(word(A)),set(word(A)),set2(word(A)),Xs))
         => ( insort_insert_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A)),Xa,Xs) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),insort_key(word(A),word(A),word_sle(A),aTP_Lamp_aad(word(A),word(A))),Xa),Xs) ) ) ) ).

% signed.insort_insert_insort
tff(fact_7754_signed_Oinsort__insert__key__triv,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xa: B,Xs: list(B)] :
          ( aa(set(word(A)),$o,member(word(A),aa(B,word(A),F,Xa)),aa(set(B),set(word(A)),image(B,word(A),F),aa(list(B),set(B),set2(B),Xs)))
         => ( insort_insert_key(word(A),B,word_sle(A),F,Xa,Xs) = Xs ) ) ) ).

% signed.insort_insert_key_triv
tff(fact_7755_signed_Osorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F: fun(B,word(A)),Xs: list(B),Xa: B] :
          ( sorted_wrt(word(A),word_sle(A),map(B,word(A),F,Xs))
         => sorted_wrt(word(A),word_sle(A),map(B,word(A),F,insort_insert_key(word(A),B,word_sle(A),F,Xa,Xs))) ) ) ).

% signed.sorted_insort_insert_key
tff(fact_7756_signed_Oinsort__insert__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F: fun(A,word(B)),Xa: A,Xs: list(A)] :
          insort_insert_key(word(B),A,word_sle(B),F,Xa,Xs) = $ite(aa(set(word(B)),$o,member(word(B),aa(A,word(B),F,Xa)),aa(set(A),set(word(B)),image(A,word(B),F),aa(list(A),set(A),set2(A),Xs))),Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insort_key(word(B),A,word_sle(B),F),Xa),Xs)) ) ).

% signed.insort_insert_key_def
tff(fact_7757_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),aa(word(B),word(A),revcast(B,A),W)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$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_7758_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F: fun(nat,set(A)),S3: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F,I2)),S3)
     => ( aa(set(A),$o,finite_finite2(A),S3)
       => ( ? [N9: nat] :
              ( ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N9)
                 => ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N9)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F,M2)),aa(nat,set(A),F,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
                 => ( aa(nat,set(A),F,N9) = aa(nat,set(A),F,N) ) ) )
         => ( aa(nat,set(A),F,aa(set(A),nat,finite_card(A),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_7759_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_as(nat,fun(nat,$o)),Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_7760_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_atMost(nat),U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_7761_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ar(nat,fun(nat,$o)),Nb))) = aa(nat,nat,suc,Nb) ).

% card_Collect_le_nat
tff(fact_7762_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_7763_card_Oinfinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_7764_card__ge__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S3: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(A),nat,finite_card(A),S3))
        <=> ( S3 = top_top(set(A)) ) ) ) ).

% card_ge_UNIV
tff(fact_7765_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),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_7766_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)),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) = aa(set(A),nat,finite_card(A),A3) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_7767_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_aaj(A,fun(B,A),Y)),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(set(B),nat,finite_card(B),A3)) ) ).

% prod_constant
tff(fact_7768_signed_Osorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] : aa(list(word(A)),nat,size_size(list(word(A))),sorted_list_of_set(word(A),word_sle(A),A3)) = aa(set(word(A)),nat,finite_card(word(A)),A3) ) ).

% signed.sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_7769_card__0__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_7770_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ~ aa(set(A),$o,member(A,Xa),A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xa),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_7771_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ie(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),aa(set(B),nat,finite_card(B),A3))),Y) ) ).

% sum_constant
tff(fact_7772_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_7773_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_aak(fun(A,$o),fun(A,B),P)),A3) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_7774_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = finite_fold(A,nat,aTP_Lamp_bl(A,fun(nat,nat)),zero_zero(nat),A3) ).

% card.eq_fold
tff(fact_7775_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S2: set(A),Ta: set(B),R: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),Ta)
       => ( ! [X: B] :
              ( aa(set(B),$o,member(B,X),Ta)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_aw(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S2),R),X))) = aa(B,nat,K,X) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_aal(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Ta),R)),S2) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7776_card__eq__sum,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_aam(A,nat)),A3) ).

% card_eq_sum
tff(fact_7777_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ).

% card_Union_le_sum_card
tff(fact_7778_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X),U2)
         => aa(set(A),$o,finite_finite2(A),X) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ) ).

% card_Union_le_sum_card_weak
tff(fact_7779_card__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),C3)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))
       => ( ! [C5: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),C5),C3)
             => ( aa(set(A),nat,finite_card(A),C5) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),C1),C3)
               => ( aa(set(set(A)),$o,member(set(A),C22),C3)
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C3)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) ) ) ) ) ) ).

% card_partition
tff(fact_7780_card__eq__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% card_eq_0_iff
tff(fact_7781_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xa),A3)) = $ite(aa(set(A),$o,member(A,Xa),A3),aa(set(A),nat,finite_card(A),A3),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_insert_if
tff(fact_7782_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B5: A,B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B5),B8) )
          & ~ aa(set(A),$o,member(A,B5),B8)
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & aa(set(A),$o,finite_finite2(A),B8) ) ) ).

% card_Suc_eq_finite
tff(fact_7783_card__image__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F),A3))),aa(set(A),nat,finite_card(A),A3)) ) ).

% card_image_le
tff(fact_7784_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) )
     => ~ ! [X: A] : A3 != aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_7785_card__insert__le,axiom,
    ! [A: $tType,A3: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xa),A3))) ).

% card_insert_le
tff(fact_7786_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),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_7787_card__Un__le,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))) ).

% card_Un_le
tff(fact_7788_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(Nb),K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_aan(nat,fun(nat,fun(set(nat),$o)),Nb),K))) ).

% binomial_def
tff(fact_7789_ucast__rev__revcast,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)),word_reverse(B,W)) = word_reverse(A,aa(word(B),word(A),revcast(B,A),W)) ) ).

% ucast_rev_revcast
tff(fact_7790_revcast__rev__ucast,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),revcast(B,A),word_reverse(B,W)) = word_reverse(A,aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)) ) ).

% revcast_rev_ucast
tff(fact_7791_ucast__revcast,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) = word_reverse(A,aa(word(B),word(A),revcast(B,A),word_reverse(B,W))) ) ).

% ucast_revcast
tff(fact_7792_revcast__ucast,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),revcast(B,A),W) = word_reverse(A,aa(word(B),word(A),semiring_1_unsigned(B,word(A)),word_reverse(B,W))) ) ).

% revcast_ucast
tff(fact_7793_distinct__card,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% distinct_card
tff(fact_7794_card__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(set(A),nat,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_7795_psubset__card__mono,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ).

% psubset_card_mono
tff(fact_7796_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bj(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_7797_n__subsets,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(nat,fun(set(A),$o),aTP_Lamp_aao(set(A),fun(nat,fun(set(A),$o)),A3),K))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A3)),K) ) ) ).

% n_subsets
tff(fact_7798_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ? [B10: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B10)
          & ( aa(set(A),nat,finite_card(A),B10) = Nb )
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B10),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_7799_card__subset__eq,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,finite_card(A),B4) )
         => ( A3 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_7800_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B4: set(A),A3: set(B),R2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( ! [A4: B] :
            ( aa(set(B),$o,member(B,A4),A3)
           => ? [B13: A] :
                ( aa(set(A),$o,member(A,B13),B4)
                & aa(A,$o,aa(B,fun(A,$o),R2,A4),B13) ) )
       => ( ! [A13: B,A24: B,B3: A] :
              ( aa(set(B),$o,member(B,A13),A3)
             => ( aa(set(B),$o,member(B,A24),A3)
               => ( aa(set(A),$o,member(A,B3),B4)
                 => ( aa(A,$o,aa(B,fun(A,$o),R2,A13),B3)
                   => ( aa(A,$o,aa(B,fun(A,$o),R2,A24),B3)
                     => ( A13 = A24 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_7801_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C3: nat] :
      ( ! [G2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G2),F4)
         => ( aa(set(A),$o,finite_finite2(A),G2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G2)),C3) ) )
     => ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_7802_card__seteq,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B4)),aa(set(A),nat,finite_card(A),A3))
         => ( A3 = B4 ) ) ) ) ).

% card_seteq
tff(fact_7803_card__mono,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ).

% card_mono
tff(fact_7804_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Nb: nat,S3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),S3))
     => ~ ! [T4: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T4),S3)
           => ( ( aa(set(A),nat,finite_card(A),T4) = Nb )
             => ~ aa(set(A),$o,finite_finite2(A),T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_7805_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
     => aa(set(A),$o,finite_finite2(A),A3) ) ).

% card_ge_0_finite
tff(fact_7806_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = nat2(U) ).

% card_atLeastZeroLessThan_int
tff(fact_7807_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,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B4),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_7808_card__le__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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B4),A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_7809_card__less__Suc2,axiom,
    ! [M6: set(nat),I: nat] :
      ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M6)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aap(set(nat),fun(nat,fun(nat,$o)),M6),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aaq(set(nat),fun(nat,fun(nat,$o)),M6),I))) ) ) ).

% card_less_Suc2
tff(fact_7810_card__less__Suc,axiom,
    ! [M6: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M6)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aap(set(nat),fun(nat,fun(nat,$o)),M6),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aaq(set(nat),fun(nat,fun(nat,$o)),M6),I))) ) ) ).

% card_less_Suc
tff(fact_7811_card__less,axiom,
    ! [M6: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M6)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aaq(set(nat),fun(nat,fun(nat,$o)),M6),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_7812_card__map__elide,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(word(A)),nat,finite_card(word(A)),top_top(set(word(A)))))
         => ( aa(set(word(A)),nat,finite_card(word(A)),aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) = aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ) ).

% card_map_elide
tff(fact_7813_revcast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),slice1(B,A,aa(word(A),nat,size_size(word(A)),aa(word(B),word(A),revcast(B,A),W))),W) = aa(word(B),word(A),revcast(B,A),W) ) ).

% revcast_slice1
tff(fact_7814_subset__card__intvl__is__intvl,axiom,
    ! [A3: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A3),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))))
     => ( A3 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_7815_sum__Suc,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_id(fun(A,nat),fun(A,nat),F)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3)),aa(set(A),nat,finite_card(A),A3)) ).

% sum_Suc
tff(fact_7816_sum__constant__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_aar(A,fun(B,A),Y)),A3) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,semiring_1_of_nat(real),aa(set(B),nat,finite_card(B),A3))),Y) ) ).

% sum_constant_scaleR
tff(fact_7817_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S3: set(A),T2: set(B),R: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ! [X: B] :
              ( aa(set(B),$o,member(B,X),T2)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_aw(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S3),R),X))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_aal(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T2),R)),S3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T2)) ) ) ) ) ).

% sum_multicount
tff(fact_7818_real__of__card,axiom,
    ! [A: $tType,A3: set(A)] : aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),A3)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_aas(A,real)),A3) ).

% real_of_card
tff(fact_7819_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))))
     => ( ( aa(set(B),nat,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_7820_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),K4) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K4)) ) ) ).

% sum_bounded_above
tff(fact_7821_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),K4: B,F: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K4),aa(A,B,F,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)) ) ) ).

% sum_bounded_below
tff(fact_7822_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
    <=> ( ( A3 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% card_gt_0_iff
tff(fact_7823_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X3: A] :
            ( aa(set(A),$o,member(A,X3),A3)
           => ! [Xa3: A] :
                ( aa(set(A),$o,member(A,Xa3),A3)
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7824_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_7825_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B5: A,B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B5),B8) )
          & ~ aa(set(A),$o,member(A,B5),B8)
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_7826_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B3: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B3),B10) )
          & ~ aa(set(A),$o,member(A,B3),B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_7827_card__1__singleton__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X3: A] : A3 = aa(set(A),set(A),insert(A,X3),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_7828_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),A3))
    <=> ? [A5: A,B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,A5),B8) )
          & ~ aa(set(A),$o,member(A,A5),B8)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),B8))
          & aa(set(A),$o,finite_finite2(A),B8) ) ) ).

% card_le_Suc_iff
tff(fact_7829_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B),F: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),aa(set(A),set(B),image(A,B,F),A3))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B4)),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% surj_card_le
tff(fact_7830_card__1__singletonI,axiom,
    ! [A: $tType,S3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( ( aa(set(A),nat,finite_card(A),S3) = one_one(nat) )
       => ( aa(set(A),$o,member(A,Xa),S3)
         => ( S3 = aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_7831_card__Diff1__le,axiom,
    ! [A: $tType,A3: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ).

% card_Diff1_le
tff(fact_7832_card__Diff__subset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B4)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ).

% card_Diff_subset
tff(fact_7833_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B4))) ) ).

% diff_card_le_card_Diff
tff(fact_7834_card__psubset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4) ) ) ) ).

% card_psubset
tff(fact_7835_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bi(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3))),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ).

% card_lists_length_le
tff(fact_7836_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_gh(nat,fun(A,$o),Nb)))),Nb) ) ) ).

% card_roots_unity
tff(fact_7837_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N7),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N7)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_7838_card__le__Suc__Max,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S3)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S3))) ) ).

% card_le_Suc_Max
tff(fact_7839_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)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_aat(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_7840_revcast__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( revcast(A,B) = slice1(A,B,type_len0_len_of(B,type2(B))) ) ) ).

% revcast_def
tff(fact_7841_card__sum__le__nat__sum,axiom,
    ! [S3: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_kx(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S3)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_kx(nat,nat)),S3)) ).

% card_sum_le_nat_sum
tff(fact_7842_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_aau(complex,fun(nat,fun(complex,$o)),C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_7843_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_hx(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_7844_card__2__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X3: A,Y5: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X3),aa(set(A),set(A),insert(A,Y5),bot_bot(set(A)))) )
          & ( X3 != Y5 ) ) ) ).

% card_2_iff
tff(fact_7845_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,F),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_7846_card__3__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X3: A,Y5: A,Z2: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X3),aa(set(A),set(A),insert(A,Y5),aa(set(A),set(A),insert(A,Z2),bot_bot(set(A))))) )
          & ( X3 != Y5 )
          & ( Y5 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_7847_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))),aa(set(A),nat,finite_card(A),A3))
     => ( A3 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_7848_card__insert__disjoint_H,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ~ aa(set(A),$o,member(A,Xa),A3)
       => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xa),A3))),aa(nat,nat,suc,zero_zero(nat))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_insert_disjoint'
tff(fact_7849_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,Xa),A3)
       => ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_7850_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xa),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_7851_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,Xa),A3)
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_7852_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,Xa),A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Diff1_less
tff(fact_7853_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),Xa: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,member(A,Xa),A3)
       => ( aa(set(A),$o,member(A,Y),A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_Diff2_less
tff(fact_7854_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))
    <=> ( aa(set(A),$o,finite_finite2(A),A3)
        & aa(set(A),$o,member(A,Xa),A3) ) ) ).

% card_Diff1_less_iff
tff(fact_7855_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ~ ! [L2: list(A)] :
                ( sorted_wrt(A,ord_less(A),L2)
               => ( ( aa(list(A),set(A),set2(A),L2) = A3 )
                 => ( aa(list(A),nat,size_size(list(A)),L2) != aa(set(A),nat,finite_card(A),A3) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_7856_card__Un__disjoint,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(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_7857_card__Diff__singleton__if,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,Xa),A3),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),one_one(nat)),aa(set(A),nat,finite_card(A),A3)) ).

% card_Diff_singleton_if
tff(fact_7858_card__Diff__singleton,axiom,
    ! [A: $tType,Xa: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,Xa),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xa),bot_bot(set(A))))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_7859_dvd__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))
     => ( ! [X: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X),C3)
           => aa(nat,$o,dvd_dvd(nat,K),aa(set(A),nat,finite_card(A),X)) )
       => ( ! [X: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X),C3)
             => ! [Xa4: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),Xa4),C3)
                 => ( ( X != Xa4 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa4) = bot_bot(set(A)) ) ) ) )
         => aa(nat,$o,dvd_dvd(nat,K),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3))) ) ) ) ).

% dvd_partition
tff(fact_7860_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S3: set(A),F: fun(A,B),K4: real] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),S3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F,X))),K4) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),S3))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),S3))),K4)) ) ) ).

% sum_norm_bound
tff(fact_7861_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_aav(nat,fun(set(A),fun(list(A),$o)),K),A3))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_kx(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7862_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_7863_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_aaw(fun(A,set(B)),fun(A,nat),A3)),I5)) ) ).

% card_UN_le
tff(fact_7864_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A3))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_co(set(A),fun(nat,fun(list(A),$o)),A3),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_kx(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_7865_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B),Nb: B,K: nat] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),Nb) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Nb)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Nb),K)) ) ) ) ) ).

% prod_le_power
tff(fact_7866_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,I2)),K4) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K4)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_7867_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A3: set(A),F: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K4),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3)))) )
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3)),K4) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_7868_signed_Osorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ~ ! [L2: list(word(A))] :
                ( sorted_wrt(word(A),word_sless(A),L2)
               => ( ( aa(list(word(A)),set(word(A)),set2(word(A)),L2) = A3 )
                 => ( aa(list(word(A)),nat,size_size(list(word(A))),L2) != aa(set(word(A)),nat,finite_card(word(A)),A3) ) ) ) ) ) ).

% signed.sorted_list_of_set.finite_set_strict_sorted
tff(fact_7869_card__insert__le__m1,axiom,
    ! [A: $tType,Nb: nat,Y: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xa),Y))),Nb) ) ) ).

% card_insert_le_m1
tff(fact_7870_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A3 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A3) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7871_card__word__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(set(word(A)),nat,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)),Xa)) ) ).

% card_word_size
tff(fact_7872_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S3: set(A),R: set(B),G: fun(A,B),F: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),R)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S3)),R)
             => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_aax(fun(A,B),fun(fun(B,C),fun(A,C)),G),F)),S3) = aa(set(B),C,aa(fun(B,C),fun(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_aay(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G),F)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_7873_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)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_aaz(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C2)),S3) = $ite(aa(set(A),$o,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,aa(set(A),nat,finite_card(A),S3)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(set(A),nat,finite_card(A),S3))) ) ) ) ).

% prod_gen_delta
tff(fact_7874_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,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_im(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_7875_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),I5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X)) )
       => ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),I5)
             => ! [Xa4: A] :
                  ( aa(set(A),$o,member(A,Xa4),I5)
                 => ( ( X != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_aaw(fun(A,set(B)),fun(A,nat),A3)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_7876_sum__le__card__Max,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image(A,nat,F),A3)))) ) ).

% sum_le_card_Max
tff(fact_7877_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_aba(nat,fun(nat,fun(list(nat),$o)),M),N7))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_abb(nat,fun(nat,fun(list(nat),$o)),M),N7)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_abc(nat,fun(nat,fun(list(nat),$o)),M),N7)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7878_card__map__elide2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(word(A)),nat,finite_card(word(A)),top_top(set(word(A)))))
         => ( aa(set(word(A)),nat,finite_card(word(A)),aa(set(nat),set(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_7879_card__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(set(word(A)),nat,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_7880_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_aba(nat,fun(nat,fun(list(nat),$o)),M),N7))) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N7),M)),one_one(nat))),N7) ).

% card_length_sum_list
tff(fact_7881_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,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_im(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_im(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_7882_signed_Osorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),L: list(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A3)
         => ( ( sorted_wrt(word(A),word_sless(A),L)
              & ( aa(list(word(A)),set(word(A)),set2(word(A)),L) = A3 )
              & ( aa(list(word(A)),nat,size_size(list(word(A))),L) = aa(set(word(A)),nat,finite_card(word(A)),A3) ) )
          <=> ( sorted_list_of_set(word(A),word_sle(A),A3) = L ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7883_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A3))
          <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_abd(set(A),fun(fun(A,B),fun(A,$o)),A3),F)))) ) ) ) ).

% even_sum_iff
tff(fact_7884_card__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(set(numeral_bit1(A)),nat,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))),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ) ).

% card_bit1
tff(fact_7885_card__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(set(numeral_bit0(A)),nat,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))),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ) ).

% card_bit0
tff(fact_7886_card__num0,axiom,
    aa(set(numeral_num0),nat,finite_card(numeral_num0),top_top(set(numeral_num0))) = zero_zero(nat) ).

% card_num0
tff(fact_7887_bit0_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size0
tff(fact_7888_bit1_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size0
tff(fact_7889_bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),Xa: numeral_bit1(A)] :
          ( ! [Z3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A))))))
               => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z3)) ) )
         => aa(numeral_bit1(A),$o,P,Xa) ) ) ).

% bit1_induct
tff(fact_7890_bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit0(A),$o),Xa: numeral_bit0(A)] :
          ( ! [Z3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A))))))
               => aa(numeral_bit0(A),$o,P,aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z3)) ) )
         => aa(numeral_bit0(A),$o,P,Xa) ) ) ).

% bit0_induct
tff(fact_7891_bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit1(A)] :
          ~ ! [Z3: int] :
              ( ( Xa = aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z3) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))))) ) ) ) ).

% bit1_cases
tff(fact_7892_bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit0(A)] :
          ~ ! [Z3: int] :
              ( ( Xa = aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z3) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))) ) ) ) ).

% bit0_cases
tff(fact_7893_card__option,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(set(option(A)),nat,finite_card(option(A)),top_top(set(option(A)))) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ) ).

% card_option
tff(fact_7894_card__prod,axiom,
    ! [A: $tType,B: $tType] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),top_top(set(product_prod(A,B)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(B),nat,finite_card(B),top_top(set(B)))) ).

% card_prod
tff(fact_7895_card__nat,axiom,
    aa(set(nat),nat,finite_card(nat),top_top(set(nat))) = zero_zero(nat) ).

% card_nat
tff(fact_7896_card__literal,axiom,
    aa(set(literal),nat,finite_card(literal),top_top(set(literal))) = zero_zero(nat) ).

% card_literal
tff(fact_7897_zero__less__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% zero_less_card_finite
tff(fact_7898_card__UNIV__sum,axiom,
    ! [B: $tType,A: $tType] :
      aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),top_top(set(sum_sum(A,B)))) = $ite(
        ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) != zero_zero(nat) )
        & ( aa(set(B),nat,finite_card(B),top_top(set(B))) != zero_zero(nat) ) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(B),nat,finite_card(B),top_top(set(B)))),
        zero_zero(nat) ) ).

% card_UNIV_sum
tff(fact_7899_one__less__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% one_less_card
tff(fact_7900_one__le__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% one_le_card_finite
tff(fact_7901_two__le__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% two_le_card
tff(fact_7902_card__UNIV__option,axiom,
    ! [A: $tType] :
      aa(set(option(A)),nat,finite_card(option(A)),top_top(set(option(A)))) = $ite(aa(set(A),nat,finite_card(A),top_top(set(A))) = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),one_one(nat))) ).

% card_UNIV_option
tff(fact_7903_card__fun,axiom,
    ! [A: $tType,B: $tType] :
      aa(set(fun(A,B)),nat,finite_card(fun(A,B)),top_top(set(fun(A,B)))) = $ite(
        ( ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) != zero_zero(nat) )
          & ( aa(set(B),nat,finite_card(B),top_top(set(B))) != zero_zero(nat) ) )
        | ( aa(set(B),nat,finite_card(B),top_top(set(B))) = one_one(nat) ) ),
        aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(B),nat,finite_card(B),top_top(set(B)))),aa(set(A),nat,finite_card(A),top_top(set(A)))),
        zero_zero(nat) ) ).

% card_fun
tff(fact_7904_card__UNIV__set,axiom,
    ! [A: $tType] :
      aa(set(set(A)),nat,finite_card(set(A)),top_top(set(set(A)))) = $ite(aa(set(A),nat,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))),aa(set(A),nat,finite_card(A),top_top(set(A))))) ).

% card_UNIV_set
tff(fact_7905_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A3)
     => ( ! [X: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X),A3)
           => aa(set(A),$o,finite_finite2(A),X) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = nat2(aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_abe(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_abf(set(set(A)),fun(set(set(A)),$o),A3)))) ) ) ) ).

% card_UNION
tff(fact_7906_Abs__bit1__inject,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: int,Y: int] :
          ( aa(set(int),$o,member(int,Xa),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),aa(set(A),nat,finite_card(A),top_top(set(A))))))))
         => ( aa(set(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),aa(set(A),nat,finite_card(A),top_top(set(A))))))))
           => ( ( aa(int,numeral_bit1(A),numeral_Abs_bit1(A),Xa) = aa(int,numeral_bit1(A),numeral_Abs_bit1(A),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% Abs_bit1_inject
tff(fact_7907_INF__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [A3: set(A)] : complete_Inf_Inf(A,aa(set(A),set(A),image(A,A,aTP_Lamp_abg(A,A)),A3)) = complete_Inf_Inf(A,A3) ) ).

% INF_identity_eq
tff(fact_7908_INT__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
         => aa(set(A),$o,member(A,B2),aa(B,set(A),B4,X3)) ) ) ).

% INT_iff
tff(fact_7909_INT__I,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B2: B,B4: fun(A,set(B))] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(set(B),$o,member(B,B2),aa(A,set(B),B4,X)) )
     => aa(set(B),$o,member(B,B2),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ).

% INT_I
tff(fact_7910_Inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [A3: set(fun(B,A)),Xa: B] : aa(B,A,complete_Inf_Inf(fun(B,A),A3),Xa) = complete_Inf_Inf(A,aa(set(fun(B,A)),set(A),image(fun(B,A),A,aTP_Lamp_abh(B,fun(fun(B,A),A),Xa)),A3)) ) ).

% Inf_apply
tff(fact_7911_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( complete_Inf_Inf(A,A3) = bot_bot(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X3)
             => ? [Xa3: A] :
                  ( aa(set(A),$o,member(A,Xa3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X3) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_7912_Inf__UNIV,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( complete_Inf_Inf(A,top_top(set(A))) = bot_bot(A) ) ) ).

% Inf_UNIV
tff(fact_7913_Inf__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( complete_Inf_Inf(A,bot_bot(set(A))) = top_top(A) ) ) ).

% Inf_empty
tff(fact_7914_ccInf__empty,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ( complete_Inf_Inf(A,bot_bot(set(A))) = top_top(A) ) ) ).

% ccInf_empty
tff(fact_7915_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : complete_Inf_Inf(A,aa(set(A),set(A),insert(A,Xa),bot_bot(set(A)))) = Xa ) ).

% cInf_singleton
tff(fact_7916_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( complete_Inf_Inf(A,set_or1337092689740270186AtMost(A,Xa,Y)) = Xa ) ) ) ).

% Inf_atLeastAtMost
tff(fact_7917_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( complete_Inf_Inf(A,set_or1337092689740270186AtMost(A,Y,Xa)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_7918_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( complete_Inf_Inf(A,set_or7035219750837199246ssThan(A,Xa,Y)) = Xa ) ) ) ).

% Inf_atLeastLessThan
tff(fact_7919_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( complete_Inf_Inf(A,set_or7035219750837199246ssThan(A,Y,Xa)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_7920_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] : complete_Inf_Inf(A,aa(A,set(A),set_ord_atMost(A),Xa)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_7921_INF__top__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( top_top(A) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,B4),A3)) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
             => ( aa(B,A,B4,X3) = top_top(A) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_7922_INF__top__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,B4),A3)) = top_top(A) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
             => ( aa(B,A,B4,X3) = top_top(A) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_7923_INF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_abi(B,A)),A3)) = top_top(A) ) ).

% INF_top
tff(fact_7924_ccINF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_abj(B,A)),A3)) = top_top(A) ) ).

% ccINF_top
tff(fact_7925_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aTP_Lamp_fd(B,fun(A,B),F)),A3)) = F ) ) ) ).

% INF_const
tff(fact_7926_ccINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aTP_Lamp_fe(B,fun(A,B),F)),A3)) = F ) ) ) ).

% ccINF_const
tff(fact_7927_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),C2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aTP_Lamp_en(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cINF_const
tff(fact_7928_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),I5)
          & aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X4)) )
     => aa(set(B),$o,finite_finite2(B),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ).

% finite_INT
tff(fact_7929_INF__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [F: fun(C,fun(B,A)),A3: set(C),Xa: B] : aa(B,A,complete_Inf_Inf(fun(B,A),aa(set(C),set(fun(B,A)),image(C,fun(B,A),F),A3)),Xa) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_abk(fun(C,fun(B,A)),fun(B,fun(C,A)),F),Xa)),A3)) ) ).

% INF_apply
tff(fact_7930_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A3: set(B)] :
          ( ( complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3)) = bot_bot(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X3)
             => ? [Xa3: B] :
                  ( aa(set(B),$o,member(B,Xa3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,Xa3)),X3) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_7931_ccINF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F: fun(B,A)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),bot_bot(set(B)))) = top_top(A) ) ).

% ccINF_empty
tff(fact_7932_INT__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_ff(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(set(A)),C2) ).

% INT_constant
tff(fact_7933_INT__insert,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A2: B,A3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(B),set(B),insert(B,A2),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B4,A2)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) ).

% INT_insert
tff(fact_7934_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : uminus_uminus(set(A),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_abl(fun(B,set(A)),fun(B,set(A)),B4)),A3)) ).

% Compl_INT
tff(fact_7935_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : uminus_uminus(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_abl(fun(B,set(A)),fun(B,set(A)),B4)),A3)) ).

% Compl_UN
tff(fact_7936_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Xa)
         => ( complete_Inf_Inf(A,aa(A,set(A),set_ord_lessThan(A),Xa)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_7937_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B4: set(A),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_nc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4)) ).

% INT_simps(1)
tff(fact_7938_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_nb(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),C3)))) ).

% INT_simps(2)
tff(fact_7939_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B4: set(A),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_fu(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),minus_minus(set(A),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4)) ).

% INT_simps(3)
tff(fact_7940_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] :
      complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_abm(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) = $ite(C3 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3)))) ).

% INT_simps(4)
tff(fact_7941_INTER__UNIV__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) = top_top(set(A)) )
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
         => ( aa(B,set(A),B4,X3) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_7942_INTER__UNIV__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( top_top(set(A)) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) )
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A3)
         => ( aa(B,set(A),B4,X3) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_7943_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,B4)),A2) = top_top(A) )
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),B4)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),A2) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_7944_Inter__UNIV,axiom,
    ! [A: $tType] : complete_Inf_Inf(set(A),top_top(set(set(A)))) = bot_bot(set(A)) ).

% Inter_UNIV
tff(fact_7945_Inter__empty,axiom,
    ! [A: $tType] : complete_Inf_Inf(set(A),bot_bot(set(set(A)))) = top_top(set(A)) ).

% Inter_empty
tff(fact_7946_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_7947_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A3: set(B)] :
          complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_fx(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(A),C2) ) ).

% INF_constant
tff(fact_7948_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,A3)),complete_Inf_Inf(A,B4))),complete_Inf_Inf(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ).

% less_eq_Inf_inter
tff(fact_7949_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),complete_Inf_Inf(A,B4)) ) ) ).

% Inf_superset_mono
tff(fact_7950_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),Xa)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y5)
             => ? [X3: A] :
                  ( aa(set(A),$o,member(A,X3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y5) ) ) ) ) ).

% Inf_le_iff
tff(fact_7951_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),complete_Inf_Inf(A,A3)) ) ) ).

% Inf_greatest
tff(fact_7952_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),complete_Inf_Inf(A,A3))
        <=> ! [X3: A] :
              ( aa(set(A),$o,member(A,X3),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X3) ) ) ) ).

% le_Inf_iff
tff(fact_7953_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),Va2: A] :
          ( aa(set(A),$o,member(A,U),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),Va2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),Va2) ) ) ) ).

% Inf_lower2
tff(fact_7954_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,Xa),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),Xa) ) ) ).

% Inf_lower
tff(fact_7955_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( ! [B3: A] :
              ( aa(set(A),$o,member(A,B3),B4)
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B3) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),complete_Inf_Inf(A,B4)) ) ) ).

% Inf_mono
tff(fact_7956_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Xa: A] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),I2) )
         => ( ! [Y3: A] :
                ( ! [I4: A] :
                    ( aa(set(A),$o,member(A,I4),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),I4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
           => ( complete_Inf_Inf(A,A3) = Xa ) ) ) ) ).

% Inf_eqI
tff(fact_7957_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X: A] :
              ( aa(set(A),$o,member(A,X),X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X) )
         => ( ! [Y3: A] :
                ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
           => ( complete_Inf_Inf(A,X5) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_7958_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Z),X5)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X) )
           => ( complete_Inf_Inf(A,X5) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_7959_Inter__lower,axiom,
    ! [A: $tType,B4: set(A),A3: set(set(A))] :
      ( aa(set(set(A)),$o,member(set(A),B4),A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),complete_Inf_Inf(set(A),A3)),B4) ) ).

% Inter_lower
tff(fact_7960_Inter__greatest,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X7),A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),X7) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),complete_Inf_Inf(set(A),A3)) ) ).

% Inter_greatest
tff(fact_7961_Inter__anti__mono,axiom,
    ! [A: $tType,B4: set(set(A)),A3: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),B4),A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),complete_Inf_Inf(set(A),A3)),complete_Inf_Inf(set(A),B4)) ) ).

% Inter_anti_mono
tff(fact_7962_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S3: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),complete_Inf_Inf(A,S3)),A2)
        <=> ? [X3: A] :
              ( aa(set(A),$o,member(A,X3),S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),A2) ) ) ) ).

% Inf_less_iff
tff(fact_7963_Inf__option__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(option(A))] :
          complete_Inf_Inf(option(A),A3) = $ite(aa(set(option(A)),$o,member(option(A),none(A)),A3),none(A),aa(A,option(A),some(A),complete_Inf_Inf(A,these(A,A3)))) ) ).

% Inf_option_def
tff(fact_7964_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),complete_Inf_Inf(A,S3))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_7965_Inter__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(A)] :
      ( ! [X7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X7),A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B4) )
     => ( ( A3 != bot_bot(set(set(A))) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),complete_Inf_Inf(set(A),A3)),B4) ) ) ).

% Inter_subset
tff(fact_7966_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),complete_Inf_Inf(A,X5)),Z)
           => ? [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z) ) ) ) ) ).

% cInf_lessD
tff(fact_7967_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X) )
           => ( ! [Y3: A] :
                  ( ! [X4: A] :
                      ( aa(set(A),$o,member(A,X4),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X4) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
             => ( complete_Inf_Inf(A,X5) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_7968_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),complete_Inf_Inf(A,X5)) ) ) ) ).

% cInf_greatest
tff(fact_7969_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V2: A] :
              ( aa(set(A),$o,member(A,V2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U) )
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),U) ) ) ) ).

% Inf_less_eq
tff(fact_7970_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Inf_le_Sup
tff(fact_7971_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A3: set(B),B4: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,M6),A3))),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,M6),B4))) ) ).

% INF_union
tff(fact_7972_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A2: B,A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),insert(B,A2),A3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,A2)),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3))) ) ).

% INF_insert
tff(fact_7973_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_abn(fun(A,B),fun(B,fun(A,B)),F),Xa)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),I5))),Xa) ) ) ) ).

% INF_inf_const2
tff(fact_7974_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),Xa: B,F: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abo(B,fun(fun(A,B),fun(A,B)),Xa),F)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Xa),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_7975_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B4: fun(B,A),A3: set(B)] : uminus_uminus(A,complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,B4),A3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_abp(fun(B,A),fun(B,A),B4)),A3)) ) ).

% uminus_INF
tff(fact_7976_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B4: fun(B,A),A3: set(B)] : uminus_uminus(A,aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B4),A3))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_abp(fun(B,A),fun(B,A),B4)),A3)) ) ).

% uminus_SUP
tff(fact_7977_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: set(A),A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),B4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3))),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,G),B4))) ) ) ) ).

% INF_superset_mono
tff(fact_7978_Int__Inter__image,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),C3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_abq(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) ).

% Int_Inter_image
tff(fact_7979_INT__Int__distrib,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),I5: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_abq(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),I5))) ).

% INT_Int_distrib
tff(fact_7980_INT__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,K)),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),A3),I5)) ) ) ).

% INT_absorb
tff(fact_7981_INF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3))),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,G),A3))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_abr(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A3)) ) ).

% INF_inf_distrib
tff(fact_7982_INF__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I5: set(A),A3: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I5)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,A3,K)),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,A3),I5))) = complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,A3),I5)) ) ) ) ).

% INF_absorb
tff(fact_7983_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => ( aa(set(A),$o,member(A,Y3),A3)
                   => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y3)),A3) ) )
             => aa(set(A),$o,member(A,complete_Inf_Inf(A,A3)),A3) ) ) ) ) ).

% finite_Inf_in
tff(fact_7984_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),complete_Inf_Inf(A,X5))
            <=> ! [X3: A] :
                  ( aa(set(A),$o,member(A,X3),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X3) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_7985_Un__Inter,axiom,
    ! [A: $tType,A3: set(A),B4: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),complete_Inf_Inf(set(A),B4)) = complete_Inf_Inf(set(A),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3)),B4)) ).

% Un_Inter
tff(fact_7986_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,B4: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),complete_Inf_Inf(A,B4)) = complete_Inf_Inf(A,aa(set(A),set(A),image(A,A,aa(A,fun(A,A),sup_sup(A),A2)),B4)) ) ).

% sup_Inf
tff(fact_7987_Pow__INT__eq,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : pow(A,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) = complete_Inf_Inf(set(set(A)),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_pi(fun(B,set(A)),fun(B,set(set(A))),B4)),A3)) ).

% Pow_INT_eq
tff(fact_7988_Inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A3: set(fun(A,B)),X4: A] : aa(A,B,complete_Inf_Inf(fun(A,B),A3),X4) = complete_Inf_Inf(B,aa(set(fun(A,B)),set(B),image(fun(A,B),B,aTP_Lamp_abs(A,fun(fun(A,B),B),X4)),A3)) ) ).

% Inf_fun_def
tff(fact_7989_INF__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aTP_Lamp_abt(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),B4)),A3)) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(set(B),fun(C,A),aTP_Lamp_abu(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A3)),B4)) ) ).

% INF_commute
tff(fact_7990_INT__D,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B),A2: B] :
      ( aa(set(A),$o,member(A,B2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
     => ( aa(set(B),$o,member(B,A2),A3)
       => aa(set(A),$o,member(A,B2),aa(B,set(A),B4,A2)) ) ) ).

% INT_D
tff(fact_7991_INT__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B),A2: B] :
      ( aa(set(A),$o,member(A,B2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
     => ( ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B4,A2))
       => ~ aa(set(B),$o,member(B,A2),A3) ) ) ).

% INT_E
tff(fact_7992_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),A3: set(B),G: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3))),complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,G),B4))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_abw(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B4)),A3)) ) ).

% INF_sup_distrib2
tff(fact_7993_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F: fun(B,A),B4: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),B4))) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_abx(A,fun(fun(B,A),fun(B,A)),A2),F)),B4)) ) ).

% sup_INF
tff(fact_7994_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,B4)),A2) = complete_Inf_Inf(A,aa(set(A),set(A),image(A,A,aTP_Lamp_aby(A,fun(A,A),A2)),B4)) ) ).

% Inf_sup
tff(fact_7995_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),B4: set(B),A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),B4))),A2) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_abz(fun(B,A),fun(A,fun(B,A)),F),A2)),B4)) ) ).

% INF_sup
tff(fact_7996_INT__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,set(A)),F: fun(B,C),A3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_fq(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B4),F)),A3)) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),B4),aa(set(B),set(C),image(B,C,F),A3))) ).

% INT_extend_simps(10)
tff(fact_7997_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),Xa: B,F: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),aa(A,B,F,I2)) )
         => ( ! [Y3: B] :
                ( ! [I4: A] :
                    ( aa(set(A),$o,member(A,I4),A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),aa(A,B,F,I4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),Xa) )
           => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3)) = Xa ) ) ) ) ).

% INF_eqI
tff(fact_7998_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B4: set(A),A3: set(B),F: fun(B,C),G: fun(A,C)] :
          ( ! [M2: A] :
              ( aa(set(A),$o,member(A,M2),B4)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),A3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X4)),aa(A,C,G,M2)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),complete_Inf_Inf(C,aa(set(B),set(C),image(B,C,F),A3))),complete_Inf_Inf(C,aa(set(A),set(C),image(A,C,G),B4))) ) ) ).

% INF_mono
tff(fact_7999_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),F: fun(A,B)] :
          ( aa(set(A),$o,member(A,I),A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3))),aa(A,B,F,I)) ) ) ).

% INF_lower
tff(fact_8000_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3))),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% INF_mono'
tff(fact_8001_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A3: set(A),F: fun(A,B),U: B] :
          ( aa(set(A),$o,member(A,I),A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3))),U) ) ) ) ).

% INF_lower2
tff(fact_8002_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3)))
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F,X3)) ) ) ) ).

% le_INF_iff
tff(fact_8003_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),U: B,F: fun(A,B)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3))) ) ) ).

% INF_greatest
tff(fact_8004_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B4: set(B),G: fun(B,C),F: fun(A,C)] :
          ( ! [I2: A] :
              ( aa(set(A),$o,member(A,I2),A3)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,X4)),aa(A,C,F,I2)) ) )
         => ( ! [J2: B] :
                ( aa(set(B),$o,member(B,J2),B4)
               => ? [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F,X4)),aa(B,C,G,J2)) ) )
           => ( complete_Inf_Inf(C,aa(set(A),set(C),image(A,C,F),A3)) = complete_Inf_Inf(C,aa(set(B),set(C),image(B,C,G),B4)) ) ) ) ) ).

% INF_eq
tff(fact_8005_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F: fun(B,A),A3: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3)))
         => ( aa(set(B),$o,member(B,I),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F,I)) ) ) ) ).

% less_INF_D
tff(fact_8006_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A3: set(B),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3))),A2)
        <=> ? [X3: B] :
              ( aa(set(B),$o,member(B,X3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X3)),A2) ) ) ) ).

% INF_less_iff
tff(fact_8007_INT__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B4: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),insert(A,A2),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ev(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B4)),C3)) ).

% INT_extend_simps(5)
tff(fact_8008_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A3: set(A),A2: B,B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A3)
     => ( complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ft(B,fun(fun(A,set(B)),fun(A,set(B))),A2),B4)),A3)) = aa(set(B),set(B),insert(B,A2),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% INT_insert_distrib
tff(fact_8009_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A3: fun(B,set(A)),I5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(B,set(A),A3,X3)) ) ) ).

% INT_subset_iff
tff(fact_8010_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F,X)),aa(A,set(B),G,X)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),F),B4))),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),G),A3))) ) ) ).

% INT_anti_mono
tff(fact_8011_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: set(B),B4: fun(A,set(B))] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),aa(A,set(B),B4,X)) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C3),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ).

% INT_greatest
tff(fact_8012_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),B4),A3))),aa(A,set(B),B4,A2)) ) ).

% INT_lower
tff(fact_8013_INT__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: fun(C,set(A)),B4: fun(B,set(C)),A3: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_aca(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C3),B4)),A3)) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),C3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image(B,set(C),B4),A3)))) ).

% INT_extend_simps(9)
tff(fact_8014_Un__INT__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B),B4: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),B4),J4))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_acc(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A3),B4),J4)),I5)) ).

% Un_INT_distrib2
tff(fact_8015_Un__INT__distrib,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A3: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_eu(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A3)),I5)) ).

% Un_INT_distrib
tff(fact_8016_INT__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_et(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3)) ).

% INT_extend_simps(6)
tff(fact_8017_INT__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_eu(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) ).

% INT_extend_simps(7)
tff(fact_8018_INF__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => ( aa(A,B,F,I2) = Xa ) )
           => ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),I5)) = Xa ) ) ) ) ).

% INF_eq_const
tff(fact_8019_Some__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(A,option(A),some(A),complete_Inf_Inf(A,A3)) = complete_Inf_Inf(option(A),aa(set(A),set(option(A)),image(A,option(A),some(A)),A3)) ) ).

% Some_Inf
tff(fact_8020_Some__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,option(A),some(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3))) = complete_Inf_Inf(option(A),aa(set(B),set(option(A)),image(B,option(A),aTP_Lamp_acd(fun(B,A),fun(B,option(A)),F)),A3)) ) ).

% Some_INF
tff(fact_8021_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A3: set(B),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,F),A3))),Xa)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y5)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X3)),Y5) ) ) ) ) ).

% INF_le_iff
tff(fact_8022_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F: fun(A,B),C2: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( aa(set(A),$o,member(A,I2),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I2)),C2) )
           => ( ( complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),I5)) = C2 )
            <=> ! [X3: A] :
                  ( aa(set(A),$o,member(A,X3),I5)
                 => ( aa(A,B,F,X3) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_8023_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),M: B,F: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(A,B,F,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3))) ) ) ) ).

% cINF_greatest
tff(fact_8024_zero__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( zero_zero(numeral_bit1(A)) = aa(int,numeral_bit1(A),numeral_Abs_bit1(A),zero_zero(int)) ) ) ).

% zero_bit1_def
tff(fact_8025_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Xa: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,Xa),X5)
           => ( ! [X: A] :
                  ( aa(set(A),$o,member(A,X),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),complete_Inf_Inf(A,X5)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_8026_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( aa(set(A),$o,member(A,Xa),X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_Inf_Inf(A,X5)),Xa) ) ) ) ).

% cInf_le_finite
tff(fact_8027_INT__empty,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A))] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_8028_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = $ite(C3 = bot_bot(set(B)),A3,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_nb(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3))) ).

% INT_extend_simps(2)
tff(fact_8029_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4) = $ite(C3 = bot_bot(set(B)),B4,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_nc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3))) ).

% INT_extend_simps(1)
tff(fact_8030_INT__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A3: set(B),B4: set(B)] : complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),M6),A3))),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),M6),B4))) ).

% INT_Un
tff(fact_8031_UN__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] : aa(set(A),set(A),minus_minus(set(A),A3),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_abm(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3)) ).

% UN_extend_simps(7)
tff(fact_8032_Inter__Un__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),complete_Inf_Inf(set(A),A3)),complete_Inf_Inf(set(A),B4))),complete_Inf_Inf(set(A),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B4))) ).

% Inter_Un_subset
tff(fact_8033_INT__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(set(B))] : complete_Inf_Inf(set(A),aa(set(set(B)),set(set(A)),image(set(B),set(A),aTP_Lamp_ace(fun(B,set(A)),fun(set(B),set(A)),B4)),A3)) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A3))) ).

% INT_extend_simps(8)
tff(fact_8034_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,A3: set(A),B9: set(set(A))] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),complete_Inf_Inf(set(A),B9)) = $ite(B9 = bot_bot(set(set(A))),A3,complete_Inf_Inf(set(A),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B9))) ).

% Int_Inter_eq(1)
tff(fact_8035_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B9: set(set(A)),A3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Inf_Inf(set(A),B9)),A3) = $ite(B9 = bot_bot(set(set(A))),A3,complete_Inf_Inf(set(A),aa(set(set(A)),set(set(A)),image(set(A),set(A),aTP_Lamp_nd(set(A),fun(set(A),set(A)),A3)),B9))) ).

% Int_Inter_eq(2)
tff(fact_8036_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_Inf_Inf(B,aa(set(A),set(B),image(A,B,F),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F),A3))) ) ) ).

% INF_le_SUP
tff(fact_8037_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,complete_Inf_Inf(A,S3)),L))),E2) ) ) ) ).

% cInf_asclose
tff(fact_8038_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S3: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( complete_Inf_Inf(A,aa(set(A),set(A),insert(A,Xa),S3)) = $ite(S3 = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),complete_Inf_Inf(A,S3))) ) ) ) ).

% Inf_insert_finite
tff(fact_8039_INF__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: fun($o,A)] : complete_Inf_Inf(A,aa(set($o),set(A),image($o,A,A3),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa($o,A,A3,$true)),aa($o,A,A3,$false)) ) ).

% INF_UNIV_bool_expand
tff(fact_8040_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B4: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),complete_Inf_Inf(A,aa(set(nat),set(A),image(nat,A,aTP_Lamp_ga(A,fun(nat,A),B4)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B4) ) ).

% INF_nat_binary
tff(fact_8041_INT__bool__eq,axiom,
    ! [A: $tType,A3: fun($o,set(A))] : complete_Inf_Inf(set(A),aa(set($o),set(set(A)),image($o,set(A),A3),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa($o,set(A),A3,$true)),aa($o,set(A),A3,$false)) ).

% INT_bool_eq
tff(fact_8042_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C3: set(B),B4: set(A)] :
      aa(set(A),set(A),minus_minus(set(A),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),A3),C3))),B4) = $ite(C3 = bot_bot(set(B)),aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),B4),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_fu(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C3))) ).

% INT_extend_simps(3)
tff(fact_8043_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C3: set(B)] :
      aa(set(A),set(A),minus_minus(set(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C3))) = $ite(C3 = bot_bot(set(B)),A3,complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_abm(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C3))) ).

% INT_extend_simps(4)
tff(fact_8044_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aTP_Lamp_acg(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,C)),set(A),image(fun(B,C),A,aTP_Lamp_aci(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_8045_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_acj(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = complete_Inf_Inf(A,aa(set(fun(B,C)),set(A),image(fun(B,C),A,aTP_Lamp_ack(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_8046_bit1_OUNIV__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( top_top(set(numeral_bit1(A))) = aa(set(int),set(numeral_bit1(A)),image(int,numeral_bit1(A),numeral_Abs_bit1(A)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A))))))) ) ) ).

% bit1.UNIV_eq
tff(fact_8047_Abs__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit1(A)] :
          ~ ! [Y3: int] :
              ( ( Xa = aa(int,numeral_bit1(A),numeral_Abs_bit1(A),Y3) )
             => ~ aa(set(int),$o,member(int,Y3),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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))) ) ) ).

% Abs_bit1_cases
tff(fact_8048_Abs__bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),Xa: numeral_bit1(A)] :
          ( ! [Y3: int] :
              ( aa(set(int),$o,member(int,Y3),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),top_top(set(A))))))))
             => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),numeral_Abs_bit1(A),Y3)) )
         => aa(numeral_bit1(A),$o,P,Xa) ) ) ).

% Abs_bit1_induct
tff(fact_8049_Inf__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( complete_Inf_Inf(A,bot_bot(set(A))) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).

% Inf_finite_empty
tff(fact_8050_Sup__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = complete_Inf_Inf(A,top_top(set(A))) ) ) ).

% Sup_finite_empty
tff(fact_8051_INF2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A3: set(A),B4: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B4,X),B2),C2) )
     => aa(C,$o,aa(B,fun(C,$o),complete_Inf_Inf(fun(B,fun(C,$o)),aa(set(A),set(fun(B,fun(C,$o))),image(A,fun(B,fun(C,$o)),B4),A3)),B2),C2) ) ).

% INF2_I
tff(fact_8052_INF1__I,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(A,fun(B,$o)),B2: B] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),A3)
         => aa(B,$o,aa(A,fun(B,$o),B4,X),B2) )
     => aa(B,$o,complete_Inf_Inf(fun(B,$o),aa(set(A),set(fun(B,$o)),image(A,fun(B,$o),B4),A3)),B2) ) ).

% INF1_I
tff(fact_8053_subset__mset_OcInf__singleton,axiom,
    ! [A: $tType,Xa: multiset(A)] : complete_Inf_Inf(multiset(A),aa(set(multiset(A)),set(multiset(A)),insert(multiset(A),Xa),bot_bot(set(multiset(A))))) = Xa ).

% subset_mset.cInf_singleton
tff(fact_8054_subset__mset_OcINF__const,axiom,
    ! [A: $tType,B: $tType,A3: set(A),C2: multiset(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( complete_Inf_Inf(multiset(B),aa(set(A),set(multiset(B)),image(A,multiset(B),aTP_Lamp_ex(multiset(B),fun(A,multiset(B)),C2)),A3)) = C2 ) ) ).

% subset_mset.cINF_const
tff(fact_8055_None__in__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(option(A))] :
          ( aa(set(option(A)),$o,member(option(A),none(A)),A3)
         => ( complete_Inf_Inf(option(A),A3) = none(A) ) ) ) ).

% None_in_Inf
tff(fact_8056_subset__mset_OcInf__eq__minimum,axiom,
    ! [A: $tType,Z: multiset(A),X5: set(multiset(A))] :
      ( aa(set(multiset(A)),$o,member(multiset(A),Z),X5)
     => ( ! [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),X5)
           => subseteq_mset(A,Z,X) )
       => ( complete_Inf_Inf(multiset(A),X5) = Z ) ) ) ).

% subset_mset.cInf_eq_minimum
tff(fact_8057_subset__mset_OcInf__eq__non__empty,axiom,
    ! [A: $tType,X5: set(multiset(A)),A2: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),X5)
           => subseteq_mset(A,A2,X) )
       => ( ! [Y3: multiset(A)] :
              ( ! [X4: multiset(A)] :
                  ( aa(set(multiset(A)),$o,member(multiset(A),X4),X5)
                 => subseteq_mset(A,Y3,X4) )
             => subseteq_mset(A,Y3,A2) )
         => ( complete_Inf_Inf(multiset(A),X5) = A2 ) ) ) ) ).

% subset_mset.cInf_eq_non_empty
tff(fact_8058_subset__mset_OcInf__greatest,axiom,
    ! [A: $tType,X5: set(multiset(A)),Z: multiset(A)] :
      ( ( X5 != bot_bot(set(multiset(A))) )
     => ( ! [X: multiset(A)] :
            ( aa(set(multiset(A)),$o,member(multiset(A),X),X5)
           => subseteq_mset(A,Z,X) )
       => subseteq_mset(A,Z,complete_Inf_Inf(multiset(A),X5)) ) ) ).

% subset_mset.cInf_greatest
tff(fact_8059_Inf__nat__def1,axiom,
    ! [K4: set(nat)] :
      ( ( K4 != bot_bot(set(nat)) )
     => aa(set(nat),$o,member(nat,complete_Inf_Inf(nat,K4)),K4) ) ).

% Inf_nat_def1
tff(fact_8060_Inf__multiset__empty,axiom,
    ! [A: $tType] : complete_Inf_Inf(multiset(A),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).

% Inf_multiset_empty
tff(fact_8061_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o)))),S3)),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),complete_Inf_Inf(set(product_prod(A,B)),S3)) ) ).

% INF_Int_eq2
tff(fact_8062_INF1__E,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,fun(A,$o)),A3: set(B),B2: A,A2: B] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B4),A3)),B2)
     => ( ~ aa(A,$o,aa(B,fun(A,$o),B4,A2),B2)
       => ~ aa(set(B),$o,member(B,A2),A3) ) ) ).

% INF1_E
tff(fact_8063_INF1__D,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,fun(A,$o)),A3: set(B),B2: A,A2: B] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B4),A3)),B2)
     => ( aa(set(B),$o,member(B,A2),A3)
       => aa(A,$o,aa(B,fun(A,$o),B4,A2),B2) ) ) ).

% INF1_D
tff(fact_8064_INF2__E,axiom,
    ! [B: $tType,A: $tType,C: $tType,B4: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B,A2: C] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B4),A3)),B2),C2)
     => ( ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B4,A2),B2),C2)
       => ~ aa(set(C),$o,member(C,A2),A3) ) ) ).

% INF2_E
tff(fact_8065_INF2__D,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B,A2: C] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B4),A3)),B2),C2)
     => ( aa(set(C),$o,member(C,A2),A3)
       => aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B4,A2),B2),C2) ) ) ).

% INF2_D
tff(fact_8066_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F4: fun(A,filter(B))] :
      ( ! [X7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B4)
         => ( aa(set(A),$o,finite_finite2(A),X7)
           => ( complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F4),X7)) != bot_bot(filter(B)) ) ) )
     => ( complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F4),B4)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_8067_subset__mset_OcINF__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),M: multiset(B),F: fun(A,multiset(B))] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => subseteq_mset(B,M,aa(A,multiset(B),F,X)) )
       => subseteq_mset(B,M,complete_Inf_Inf(multiset(B),aa(set(A),set(multiset(B)),image(A,multiset(B),F),A3))) ) ) ).

% subset_mset.cINF_greatest
tff(fact_8068_subset__mset_OcInf__le__finite,axiom,
    ! [A: $tType,X5: set(multiset(A)),Xa: multiset(A)] :
      ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),X5)
     => ( aa(set(multiset(A)),$o,member(multiset(A),Xa),X5)
       => subseteq_mset(A,complete_Inf_Inf(multiset(A),X5),Xa) ) ) ).

% subset_mset.cInf_le_finite
tff(fact_8069_INF__Int__eq,axiom,
    ! [A: $tType,S3: set(set(A)),X4: A] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(set(A)),set(fun(A,$o)),image(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o))),S3)),X4)
    <=> aa(set(A),$o,member(A,X4),complete_Inf_Inf(set(A),S3)) ) ).

% INF_Int_eq
tff(fact_8070_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S3: set(B),X4: A] :
      ( aa(A,$o,complete_Inf_Inf(fun(A,$o),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),aTP_Lamp_fh(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S3)),X4)
    <=> aa(set(A),$o,member(A,X4),complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),R2),S3))) ) ).

% INF_INT_eq
tff(fact_8071_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S3: set(C),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_fi(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S3)),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),complete_Inf_Inf(set(product_prod(A,B)),aa(set(C),set(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R2),S3))) ) ).

% INF_INT_eq2
tff(fact_8072_Inf__set__def,axiom,
    ! [A: $tType,A3: set(set(A))] : complete_Inf_Inf(set(A),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_acl(set(set(A)),fun(A,$o),A3)) ).

% Inf_set_def
tff(fact_8073_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,$o))),X4: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),complete_Inf_Inf(fun(A,fun(B,$o)),S3),X4),Xa2)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2)),complete_Inf_Inf(set(product_prod(A,B)),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S3)))) ) ).

% Inf_INT_eq2
tff(fact_8074_bot__finite__def,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( bot_bot(A) = complete_Inf_Inf(A,top_top(set(A))) ) ) ).

% bot_finite_def
tff(fact_8075_Abs__bit1__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( aa(set(int),$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),aa(set(A),nat,finite_card(A),top_top(set(A))))))))
         => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit1(A),Y)) = Y ) ) ) ).

% Abs_bit1_inverse
tff(fact_8076_Rep__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit1(A)] : aa(set(int),$o,member(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xa)),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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))) ) ).

% Rep_bit1
tff(fact_8077_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F4: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( aa(set(A),$o,member(A,I2),I5)
         => ! [J2: A] :
              ( aa(set(A),$o,member(A,J2),I5)
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),I5)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I2)),aa(A,filter(B),F4,J2))) ) ) )
     => ( ( complete_Inf_Inf(filter(B),aa(set(A),set(filter(B)),image(A,filter(B),F4),I5)) = bot_bot(filter(B)) )
      <=> ? [X3: A] :
            ( aa(set(A),$o,member(A,X3),I5)
            & ( aa(A,filter(B),F4,X3) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_8078_Inf__filter__not__bot,axiom,
    ! [A: $tType,B4: set(filter(A))] :
      ( ! [X7: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X7),B4)
         => ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X7)
           => ( complete_Inf_Inf(filter(A),X7) != bot_bot(filter(A)) ) ) )
     => ( complete_Inf_Inf(filter(A),B4) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_8079_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_bit1(A),zero_zero(int))) = zero_zero(int) ) ) ).

% bit1.Rep_Abs_0
tff(fact_8080_less__eq__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A2: numeral_bit1(A),B2: numeral_bit1(A)] :
          ( aa(numeral_bit1(A),$o,aa(numeral_bit1(A),fun(numeral_bit1(A),$o),ord_less_eq(numeral_bit1(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),A2)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),B2)) ) ) ).

% less_eq_bit1_def
tff(fact_8081_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_8082_bit1_ORep__le__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit1(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Xa)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_le_n
tff(fact_8083_bit1_OAbs__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [M: int] :
          ( aa(set(int),$o,member(int,M),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,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_bit1(A),M)) = M ) ) ) ).

% bit1.Abs_inverse
tff(fact_8084_Rep__bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int,P: fun(int,$o)] :
          ( aa(set(int),$o,member(int,Y),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),top_top(set(A))))))))
         => ( ! [X: numeral_bit1(A)] : aa(int,$o,P,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X))
           => aa(int,$o,P,Y) ) ) ) ).

% Rep_bit1_induct
tff(fact_8085_Rep__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( aa(set(int),$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),aa(set(A),nat,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_8086_type__definition__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit1(A),int,numeral_Rep_bit1(A),numeral_Abs_bit1(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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))) ) ).

% type_definition_bit1
tff(fact_8087_Abs__bit0__inject,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: int,Y: int] :
          ( aa(set(int),$o,member(int,Xa),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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))
         => ( aa(set(int),$o,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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))
           => ( ( aa(int,numeral_bit0(A),numeral_Abs_bit0(A),Xa) = aa(int,numeral_bit0(A),numeral_Abs_bit0(A),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% Abs_bit0_inject
tff(fact_8088_less__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F4),F6)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F6)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F6),F4) ) ) ).

% less_filter_def
tff(fact_8089_zero__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( zero_zero(numeral_bit0(A)) = aa(int,numeral_bit0(A),numeral_Abs_bit0(A),zero_zero(int)) ) ) ).

% zero_bit0_def
tff(fact_8090_bit0_OUNIV__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( top_top(set(numeral_bit0(A))) = aa(set(int),set(numeral_bit0(A)),image(int,numeral_bit0(A),numeral_Abs_bit0(A)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A))))))) ) ) ).

% bit0.UNIV_eq
tff(fact_8091_bit1_Otype,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit1(A),int,numeral_Rep_bit1(A),numeral_Abs_bit1(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.type
tff(fact_8092_Abs__bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit0(A)] :
          ~ ! [Y3: int] :
              ( ( Xa = aa(int,numeral_bit0(A),numeral_Abs_bit0(A),Y3) )
             => ~ aa(set(int),$o,member(int,Y3),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),aa(set(A),nat,finite_card(A),top_top(set(A))))))) ) ) ).

% Abs_bit0_cases
tff(fact_8093_Abs__bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit0(A),$o),Xa: numeral_bit0(A)] :
          ( ! [Y3: int] :
              ( aa(set(int),$o,member(int,Y3),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),top_top(set(A)))))))
             => aa(numeral_bit0(A),$o,P,aa(int,numeral_bit0(A),numeral_Abs_bit0(A),Y3)) )
         => aa(numeral_bit0(A),$o,P,Xa) ) ) ).

% Abs_bit0_induct
tff(fact_8094_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,aa(int,fun(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)) )
            & ! [X3: A,Y5: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y5) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X3)),aa(A,int,Rep,Y5)),Nb))
            & ! [X3: A,Y5: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y5) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X3)),aa(A,int,Rep,Y5)),Nb))
            & ! [X3: A,Y5: A] : aa(A,A,minus_minus(A,X3),Y5) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,minus_minus(int,aa(A,int,Rep,X3)),aa(A,int,Rep,Y5)),Nb))
            & ! [X3: A] : uminus_uminus(A,X3) = aa(int,A,Abs,modulo_modulo(int,uminus_uminus(int,aa(A,int,Rep,X3)),Nb)) ) ) ) ).

% mod_type_def
tff(fact_8095_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,aa(int,fun(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,Y3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) = 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,Y3)),Nb))
                 => ( ! [X: A,Y3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3) = 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,Y3)),Nb))
                   => ( ! [X: A,Y3: A] : aa(A,A,minus_minus(A,X),Y3) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,minus_minus(int,aa(A,int,Rep,X)),aa(A,int,Rep,Y3)),Nb))
                     => ( ! [X: A] : uminus_uminus(A,X) = aa(int,A,Abs,modulo_modulo(int,uminus_uminus(int,aa(A,int,Rep,X)),Nb))
                       => numeral_mod_type(A,Nb,Rep,Abs) ) ) ) ) ) ) ) ) ) ).

% mod_type.intro
tff(fact_8096_mod__type_ORep__le__n,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),Xa: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,Rep,Xa)),Nb) ) ) ).

% mod_type.Rep_le_n
tff(fact_8097_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_8098_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_8099_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_8100_mod__type_OAbs__inverse,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Nb: int,Rep: fun(A,int),Abs: fun(int,A),M: int] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(set(int),$o,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_8101_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,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb) ) ) ).

% mod_type.size0
tff(fact_8102_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),Xa: A,Y: A] :
          ( numeral_mod_type(A,Nb,Rep,Abs)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(int,A,Abs,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,Xa)),aa(A,int,Rep,Y)),Nb)) ) ) ) ).

% mod_type.mult_def
tff(fact_8103_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_8104_type__definition__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit0(A),int,numeral_Rep_bit0(A),numeral_Abs_bit0(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),aa(set(A),nat,finite_card(A),top_top(set(A))))))) ) ).

% type_definition_bit0
tff(fact_8105_Abs__bit0__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( aa(set(int),$o,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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))
         => ( aa(numeral_bit0(A),int,numeral_Rep_bit0(A),aa(int,numeral_bit0(A),numeral_Abs_bit0(A),Y)) = Y ) ) ) ).

% Abs_bit0_inverse
tff(fact_8106_bit0_ORep__Abs__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit0(A),int,numeral_Rep_bit0(A),aa(int,numeral_bit0(A),numeral_Abs_bit0(A),zero_zero(int))) = zero_zero(int) ) ) ).

% bit0.Rep_Abs_0
tff(fact_8107_less__eq__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A2: numeral_bit0(A),B2: numeral_bit0(A)] :
          ( aa(numeral_bit0(A),$o,aa(numeral_bit0(A),fun(numeral_bit0(A),$o),ord_less_eq(numeral_bit0(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),A2)),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),B2)) ) ) ).

% less_eq_bit0_def
tff(fact_8108_bit0_ORep__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit0(A),int,numeral_Rep_bit0(A),zero_zero(numeral_bit0(A))) = zero_zero(int) ) ) ).

% bit0.Rep_0
tff(fact_8109_bit0_ORep__le__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit0(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Xa)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.Rep_le_n
tff(fact_8110_bit0_OAbs__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [M: int] :
          ( aa(set(int),$o,member(int,M),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))))
         => ( aa(numeral_bit0(A),int,numeral_Rep_bit0(A),aa(int,numeral_bit0(A),numeral_Abs_bit0(A),M)) = M ) ) ) ).

% bit0.Abs_inverse
tff(fact_8111_bit0_Otype,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit0(A),int,numeral_Rep_bit0(A),numeral_Abs_bit0(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.type
tff(fact_8112_Rep__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit0(A)] : aa(set(int),$o,member(int,aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Xa)),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),aa(set(A),nat,finite_card(A),top_top(set(A))))))) ) ).

% Rep_bit0
tff(fact_8113_Rep__bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( aa(set(int),$o,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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))
         => ~ ! [X: numeral_bit0(A)] : Y != aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X) ) ) ).

% Rep_bit0_cases
tff(fact_8114_Rep__bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int,P: fun(int,$o)] :
          ( aa(set(int),$o,member(int,Y),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),top_top(set(A)))))))
         => ( ! [X: numeral_bit0(A)] : aa(int,$o,P,aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X))
           => aa(int,$o,P,Y) ) ) ) ).

% Rep_bit0_induct
tff(fact_8115_inj__on__Abs__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => inj_on(int,numeral_bit1(A),numeral_Abs_bit1(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),aa(set(A),nat,finite_card(A),top_top(set(A)))))))) ) ).

% inj_on_Abs_bit1
tff(fact_8116_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),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))) = zero_zero(int) ) ) ) ).

% bit0_iszero_numeral
tff(fact_8117_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : inj_on(A,B,F,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_8118_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_8119_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_acm(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_8120_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A2: A,A3: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),insert(A,A2),A3))
    <=> ( inj_on(A,B,F,A3)
        & ~ aa(set(B),$o,member(B,aa(A,B,F,A2)),aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_8121_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( inj_on(B,A,F,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_acn(set(A),fun(fun(B,A),fun(B,$o)),A3),F))) ) ) ).

% finite_inverse_image
tff(fact_8122_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( inj_on(B,A,F,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_acn(set(A),fun(fun(B,A),fun(B,$o)),S3),F))) ) ) ).

% finite_Collect
tff(fact_8123_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(B,A),D4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( inj_on(B,A,F,D4)
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_aco(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A3),F),D4))) ) ) ).

% finite_inverse_image_gen
tff(fact_8124_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B4: set(A),A3: set(A)] :
      ( inj_on(A,B,F,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => inj_on(A,B,F,A3) ) ) ).

% subset_inj_on
tff(fact_8125_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
       => inj_on(A,B,F,B4) ) ) ).

% inj_on_subset
tff(fact_8126_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_acp(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_8127_inj__on__id2,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,A,aTP_Lamp_bp(A,A),A3) ).

% inj_on_id2
tff(fact_8128_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),insert(A,Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_acq(fun(A,B),fun(A,fun(A,$o)),F),Y)),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_8129_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_dw(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_8130_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_acr(fun(A,B),fun(A,fun(C,B)),F),top_top(set(A))) ) ).

% inj_fun
tff(fact_8131_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_eg(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_8132_inj__on__Inter,axiom,
    ! [B: $tType,A: $tType,S3: set(set(A)),F: fun(A,B)] :
      ( ( S3 != bot_bot(set(set(A))) )
     => ( ! [A7: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),A7),S3)
           => inj_on(A,B,F,A7) )
       => inj_on(A,B,F,complete_Inf_Inf(set(A),S3)) ) ) ).

% inj_on_Inter
tff(fact_8133_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F),A3))),aa(set(B),nat,finite_card(B),A3))
     => ~ inj_on(B,A,F,A3) ) ).

% pigeonhole
tff(fact_8134_sum_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),aTP_Lamp_acs(B,B)),aa(set(A),set(B),image(A,B,G),A3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) ) ) ) ).

% sum.image_eq
tff(fact_8135_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F: fun(B,C),A3: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I5)
           => inj_on(B,C,F,aa(A,set(B),A3,I2)) )
       => inj_on(B,C,F,complete_Inf_Inf(set(B),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ) ).

% inj_on_INTER
tff(fact_8136_prod_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7121269368397514597t_prod(B,B),aTP_Lamp_act(B,B)),aa(set(A),set(B),image(A,B,G),A3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) ) ) ) ).

% prod.image_eq
tff(fact_8137_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F: fun(A,B)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => ( aa(A,B,F,X) != aa(A,B,F,Y3) ) )
         => inj_on(A,B,F,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_8138_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num] :
          ( ( aa(num,A,numeral_numeral(A),Xa) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xa)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_8139_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_8140_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_8141_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_8142_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_8143_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_8144_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => ( aa(set(A),$o,member(A,X),A3)
               => ( aa(set(A),$o,member(A,Y3),A3)
                 => ( aa(A,B,F,X) != aa(A,B,F,Y3) ) ) ) )
         => ( ! [X: A,Y3: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => ( aa(set(A),$o,member(A,Y3),A3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) ) ) )
           => inj_on(A,B,F,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_8145_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C3: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F,C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
         => ( aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B4)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_8146_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F: fun(B,A),T2: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),aa(set(B),set(A),image(B,A,F),T2))
    <=> ? [U5: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),U5),T2)
          & inj_on(B,A,F,U5)
          & ( S3 = aa(set(B),set(A),image(B,A,F),U5) ) ) ) ).

% subset_image_inj
tff(fact_8147_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B4: set(A),A2: A,A3: set(A)] :
      ( inj_on(A,B,F,B4)
     => ( aa(set(A),$o,member(A,A2),B4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ( aa(set(B),$o,member(B,aa(A,B,F,A2)),aa(set(A),set(B),image(A,B,F),A3))
          <=> aa(set(A),$o,member(A,A2),A3) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_8148_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C3: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F,C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
         => ( ( aa(set(A),set(B),image(A,B,F),A3) = aa(set(A),set(B),image(A,B,F),B4) )
          <=> ( A3 = B4 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_8149_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A9: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A3)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F7),A3)),A9) )
      <=> ? [G3: fun(B,A)] : aa(set(B),set(A),image(B,A,G3),A9) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_8150_endo__inj__surj,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,F),A3)),A3)
       => ( inj_on(A,A,F,A3)
         => ( aa(set(A),set(A),image(A,A,F),A3) = A3 ) ) ) ) ).

% endo_inj_surj
tff(fact_8151_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B4: set(B)] :
      ( inj_on(A,B,F,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),B4)
       => ( aa(set(B),$o,finite_finite2(B),B4)
         => aa(set(A),$o,finite_finite2(A),A3) ) ) ) ).

% inj_on_finite
tff(fact_8152_finite__surj__inj,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),image(A,A,F),A3))
       => inj_on(A,A,F,A3) ) ) ).

% finite_surj_inj
tff(fact_8153_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F: fun(B,C)] :
      ( ! [I2: A,J2: A] :
          ( aa(set(A),$o,member(A,I2),I5)
         => ( aa(set(A),$o,member(A,J2),I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,I2)),aa(A,set(B),A3,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I2)) ) ) )
     => ( ! [I2: A] :
            ( aa(set(A),$o,member(A,I2),I5)
           => inj_on(B,C,F,aa(A,set(B),A3,I2)) )
       => inj_on(B,C,F,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_8154_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C3: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F,C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B4)),C3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C3)
         => ( aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),B4)) = aa(set(B),set(B),minus_minus(set(B),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B4)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_8155_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B4))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ) ).

% inj_image_subset_iff
tff(fact_8156_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A3: set(A),B4: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),B4)
       => ( inj_on(B,A,G,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),B4)),A3)
           => ( aa(set(A),$o,finite_finite2(A),A3)
             => ( aa(set(B),$o,finite_finite2(B),B4)
               => ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B4) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_8157_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S3: set(A),T2: set(B),F: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ( aa(set(A),nat,finite_card(A),S3) = aa(set(B),nat,finite_card(B),T2) )
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),S3)),T2)
           => ( ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),T2)
                 => ? [Xa3: A] :
                      ( aa(set(A),$o,member(A,Xa3),S3)
                      & ( aa(A,B,F,Xa3) = X3 ) ) )
            <=> inj_on(A,B,F,S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_8158_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),uminus_uminus(set(A),A3))),uminus_uminus(set(B),aa(set(A),set(B),image(A,B,F),A3))) ) ).

% inj_image_Compl_subset
tff(fact_8159_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(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_8160_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num] :
          ( ( uminus_uminus(A,aa(num,A,numeral_numeral(A),Xa)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xa)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_8161_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),G: fun(A,B),B4: set(A)] :
      ( inj_on(A,B,F,A3)
     => ( inj_on(A,B,G,B4)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,G),B4)) = bot_bot(set(B)) )
         => inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_acu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_8162_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),C3: set(A),A3: set(C),B4: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F,C3)
     => ( ! [X: C] :
            ( aa(set(C),$o,member(C,X),A3)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(C,set(A),B4,X)),C3) )
       => ( aa(set(C),$o,member(C,J),A3)
         => ( aa(set(A),set(B),image(A,B,F),complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),B4),A3))) = complete_Inf_Inf(set(B),aa(set(C),set(set(B)),image(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_acv(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F),B4)),A3)) ) ) ) ) ).

% image_INT
tff(fact_8163_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( ? [F7: fun(A,B)] :
              ( inj_on(A,B,F7,A3)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F7),A3)),B4) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_8164_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A3: set(A),B4: set(B)] :
      ( inj_on(A,B,F,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),B4)
       => ( aa(set(B),$o,finite_finite2(B),B4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% card_inj_on_le
tff(fact_8165_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4))
         => ? [F2: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4)
              & inj_on(A,B,F2,A3) ) ) ) ) ).

% card_le_inj
tff(fact_8166_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))
    <=> ( inj_on(A,B,F,A3)
        & inj_on(A,B,F,B4)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),B4))),aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),minus_minus(set(A),B4),A3))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_8167_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A),Ys: list(A)] :
          ( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
         => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F,Xs))
           => ( distinct(B,map(A,B,F,Xs))
             => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F,Ys))
               => ( distinct(B,map(A,B,F,Ys))
                 => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_8168_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),A3: set(A),F: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A3)
     => ( inj_on(A,B,F,A3)
       => ( map(B,A,inv_on(A,B,F,A3),map(A,B,F,L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_8169_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),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))))) = zero_zero(int) ) ) ) ).

% bit1_iszero_numeral
tff(fact_8170_word__eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),Xa) = zero_zero(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Xa)) ) ) ).

% word_eq_numeral_iff_iszero(9)
tff(fact_8171_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_8172_word__eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] :
          ( ( uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),Xa)) = zero_zero(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Xa)) ) ) ).

% word_eq_numeral_iff_iszero(11)
tff(fact_8173_word__eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( zero_zero(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_8174_word__eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num,Y: num] :
          ( ( uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),Xa)) = 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),Xa),Y))) ) ) ).

% word_eq_numeral_iff_iszero(3)
tff(fact_8175_word__eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num,Y: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),Xa) = 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),Xa),Y))) ) ) ).

% word_eq_numeral_iff_iszero(2)
tff(fact_8176_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_8177_word__eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( one_one(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_8178_word__eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] :
          ( ( uminus_uminus(word(A),aa(num,word(A),numeral_numeral(word(A)),Xa)) = 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),Xa),one2))) ) ) ).

% word_eq_numeral_iff_iszero(7)
tff(fact_8179_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_8180_ATP_Olambda__1,axiom,
    ! [Uu: option(nat)] : aa(option(nat),assn,aTP_Lamp_a(option(nat),assn),Uu) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),v),aa(nat,nat,suc,zero_zero(nat)),vg,vh)),tia)),pure_assn(Uu = 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))) ).

% ATP.lambda_1
tff(fact_8181_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_qs(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),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_2
tff(fact_8182_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_abe(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),complete_Inf_Inf(set(A),Uu)))) ).

% ATP.lambda_3
tff(fact_8183_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_es(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_4
tff(fact_8184_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_sm(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_5
tff(fact_8185_ATP_Olambda__6,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_wm(real,$o),Uu)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_6
tff(fact_8186_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_km(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_7
tff(fact_8187_ATP_Olambda__8,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_zf(word(B),$o),Uu)
        <=> aa(word(B),$o,aa(word(B),fun(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_8
tff(fact_8188_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_le(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_9
tff(fact_8189_ATP_Olambda__10,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_cn(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).

% ATP.lambda_10
tff(fact_8190_ATP_Olambda__11,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_dl(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_11
tff(fact_8191_ATP_Olambda__12,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,list(A),aTP_Lamp_if(A,list(A)),Uu) = aa(list(A),list(A),cons(A,Uu),nil(A)) ) ).

% ATP.lambda_12
tff(fact_8192_ATP_Olambda__13,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_hy(A,list(A)),Uu) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_13
tff(fact_8193_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_ew(A,set(A)),Uu) = aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))) ).

% ATP.lambda_14
tff(fact_8194_ATP_Olambda__15,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_lm(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),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_15
tff(fact_8195_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_dk(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_16
tff(fact_8196_ATP_Olambda__17,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_ln(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_17
tff(fact_8197_ATP_Olambda__18,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_lc(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_18
tff(fact_8198_ATP_Olambda__19,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_vn(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = aa(fun(A,fun(B,fun(A,option(B)))),fun(product_prod(A,B),fun(A,option(B))),product_case_prod(A,B,fun(A,option(B))),aTP_Lamp_vm(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_19
tff(fact_8199_ATP_Olambda__20,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_ue(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_ud(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_20
tff(fact_8200_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tp(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_21
tff(fact_8201_ATP_Olambda__22,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ll(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ).

% ATP.lambda_22
tff(fact_8202_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_jf(list(A),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu)) ).

% ATP.lambda_23
tff(fact_8203_ATP_Olambda__24,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_xh(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_24
tff(fact_8204_ATP_Olambda__25,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_xf(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).

% ATP.lambda_25
tff(fact_8205_ATP_Olambda__26,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_xz(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),uminus_uminus(int,Uu)) ).

% ATP.lambda_26
tff(fact_8206_ATP_Olambda__27,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_yc(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_27
tff(fact_8207_ATP_Olambda__28,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_we(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_28
tff(fact_8208_ATP_Olambda__29,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_xl(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_29
tff(fact_8209_ATP_Olambda__30,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_xo(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_xm(nat,fun(num,option(num)),Uua),aTP_Lamp_xn(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_30
tff(fact_8210_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_tx(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_31
tff(fact_8211_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_qp(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_32
tff(fact_8212_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_tw(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_33
tff(fact_8213_ATP_Olambda__34,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_sn(fun(nat,real),fun(nat,real),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(real),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_34
tff(fact_8214_ATP_Olambda__35,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] :
      aa(nat,option(A),aTP_Lamp_bo(list(A),fun(nat,option(A)),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(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_35
tff(fact_8215_ATP_Olambda__36,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_xq(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_36
tff(fact_8216_ATP_Olambda__37,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_qo(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_37
tff(fact_8217_ATP_Olambda__38,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_wx(option(A),fun(A,option(A)),Uu),Uua) = case_option(option(A),A,none(A),aTP_Lamp_ww(A,fun(A,option(A)),Uua),Uu) ) ).

% ATP.lambda_38
tff(fact_8218_ATP_Olambda__39,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_xc(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$true,aTP_Lamp_xb(A,fun(A,$o),Uua),Uu) ) ) ).

% ATP.lambda_39
tff(fact_8219_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_xp(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_xo(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_40
tff(fact_8220_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_xa(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$false,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_41
tff(fact_8221_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_xm(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_xf(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_42
tff(fact_8222_ATP_Olambda__43,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_xj(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_xf(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_43
tff(fact_8223_ATP_Olambda__44,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_ku(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kt(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_44
tff(fact_8224_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_vd(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_vc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_45
tff(fact_8225_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_vb(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_va(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_46
tff(fact_8226_ATP_Olambda__47,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qd(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),uminus_uminus(real,one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_47
tff(fact_8227_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sd(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),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_48
tff(fact_8228_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sp(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),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_49
tff(fact_8229_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_mn(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),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_50
tff(fact_8230_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_pq(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_51
tff(fact_8231_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_wi(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_52
tff(fact_8232_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_wh(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( tan(real,Uua) = Uu ) ) ) ).

% ATP.lambda_53
tff(fact_8233_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_vy(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_54
tff(fact_8234_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_wp(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_55
tff(fact_8235_ATP_Olambda__56,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_wq(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_56
tff(fact_8236_ATP_Olambda__57,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_cq(set(A),fun(list(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)
        & distinct(A,Uua) ) ) ).

% ATP.lambda_57
tff(fact_8237_ATP_Olambda__58,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_cp(set(A),fun(list(A),$o),Uu),Uua)
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
        & distinct(A,Uua) ) ) ).

% ATP.lambda_58
tff(fact_8238_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qe(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),uminus_uminus(real,one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_59
tff(fact_8239_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_me(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),uminus_uminus(A,one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_60
tff(fact_8240_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lw(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_61
tff(fact_8241_ATP_Olambda__62,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_pu(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_62
tff(fact_8242_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lx(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),uminus_uminus(A,one_one(A))),Uua)) ) ).

% ATP.lambda_63
tff(fact_8243_ATP_Olambda__64,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_abf(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_64
tff(fact_8244_ATP_Olambda__65,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_gc(set(option(A)),fun(option(A),$o),Uu),Uua)
    <=> ( aa(set(option(A)),$o,member(option(A),Uua),Uu)
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_65
tff(fact_8245_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_po(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_66
tff(fact_8246_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ks(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_67
tff(fact_8247_ATP_Olambda__68,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_gf(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_68
tff(fact_8248_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mh(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_69
tff(fact_8249_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mg(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_70
tff(fact_8250_ATP_Olambda__71,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_el(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_71
tff(fact_8251_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_hx(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_72
tff(fact_8252_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_gh(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_73
tff(fact_8253_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qb(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_74
tff(fact_8254_ATP_Olambda__75,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_wr(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi)
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_75
tff(fact_8255_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_pn(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_76
tff(fact_8256_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_pm(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_77
tff(fact_8257_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ho(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_78
tff(fact_8258_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_oq(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_79
tff(fact_8259_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jl(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_80
tff(fact_8260_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rt(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_8261_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qg(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_82
tff(fact_8262_ATP_Olambda__83,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gp(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_83
tff(fact_8263_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ru(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_84
tff(fact_8264_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jv(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_85
tff(fact_8265_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ju(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_86
tff(fact_8266_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ij(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_87
tff(fact_8267_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ou(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_88
tff(fact_8268_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kg(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_89
tff(fact_8269_ATP_Olambda__90,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_sc(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_90
tff(fact_8270_ATP_Olambda__91,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_go(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_8271_ATP_Olambda__92,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_fz(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),insert(A,aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_92
tff(fact_8272_ATP_Olambda__93,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_hm(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_93
tff(fact_8273_ATP_Olambda__94,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ok(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_94
tff(fact_8274_ATP_Olambda__95,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_dj(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aTP_Lamp_di(list(A),fun(A,list(A)),Uua),Uu) ).

% ATP.lambda_95
tff(fact_8275_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_vg(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).

% ATP.lambda_96
tff(fact_8276_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_to(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_97
tff(fact_8277_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tu(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_98
tff(fact_8278_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tr(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_99
tff(fact_8279_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tq(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_100
tff(fact_8280_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_sq(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_101
tff(fact_8281_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: num,Uua: num] : aa(num,word(A),aTP_Lamp_xe(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_102
tff(fact_8282_ATP_Olambda__103,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_xg(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_103
tff(fact_8283_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tg(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),uminus_uminus(A,Uu)),Uua)) ) ).

% ATP.lambda_104
tff(fact_8284_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tb(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_105
tff(fact_8285_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tc(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_106
tff(fact_8286_ATP_Olambda__107,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qu(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_107
tff(fact_8287_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qt(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_108
tff(fact_8288_ATP_Olambda__109,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_dg(list(list(A)),fun(A,list(list(A))),Uu),Uua) = map(list(A),list(A),cons(A,Uua),product_lists(A,Uu)) ).

% ATP.lambda_109
tff(fact_8289_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ly(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_110
tff(fact_8290_ATP_Olambda__111,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_bz(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_111
tff(fact_8291_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_wj(real,fun(real,$o),Uu),Uua)
    <=> ( exp(real,Uua) = Uu ) ) ).

% ATP.lambda_112
tff(fact_8292_ATP_Olambda__113,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_ub(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_113
tff(fact_8293_ATP_Olambda__114,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_uc(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_114
tff(fact_8294_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_px(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_115
tff(fact_8295_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_pg(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,Uu)),Uua)) ).

% ATP.lambda_116
tff(fact_8296_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_wc(list(A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% ATP.lambda_117
tff(fact_8297_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_wd(list(A),fun(A,$o),Uu),Uua)
        <=> ( Uua = aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% ATP.lambda_118
tff(fact_8298_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ps(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_119
tff(fact_8299_ATP_Olambda__120,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_wl(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_120
tff(fact_8300_ATP_Olambda__121,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_lh(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu,Uua)) ).

% ATP.lambda_121
tff(fact_8301_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_yo(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_122
tff(fact_8302_ATP_Olambda__123,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_vl(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_123
tff(fact_8303_ATP_Olambda__124,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_zl(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ).

% ATP.lambda_124
tff(fact_8304_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_zh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_125
tff(fact_8305_ATP_Olambda__126,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_kq(nat,fun(list(A),$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_126
tff(fact_8306_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_nl(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_127
tff(fact_8307_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_pd(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_128
tff(fact_8308_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_oz(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_129
tff(fact_8309_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_eq(list(set(A)),fun(set(A),$o),Uu),Uua)
    <=> aa(set(set(A)),$o,member(set(A),Uua),aa(list(set(A)),set(set(A)),set2(set(A)),Uu)) ) ).

% ATP.lambda_130
tff(fact_8310_ATP_Olambda__131,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_at(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).

% ATP.lambda_131
tff(fact_8311_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ar(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_132
tff(fact_8312_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_du(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_133
tff(fact_8313_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_gd(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_134
tff(fact_8314_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ii(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_135
tff(fact_8315_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zk(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_136
tff(fact_8316_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ji(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_137
tff(fact_8317_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_acm(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_138
tff(fact_8318_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_as(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_8319_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_dv(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_140
tff(fact_8320_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_xb(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_141
tff(fact_8321_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ka(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_142
tff(fact_8322_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_dx(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_143
tff(fact_8323_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_jp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_8324_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dy(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_145
tff(fact_8325_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_jh(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_8326_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_eg(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_147
tff(fact_8327_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aby(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_148
tff(fact_8328_ATP_Olambda__149,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_nd(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).

% ATP.lambda_149
tff(fact_8329_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_150
tff(fact_8330_ATP_Olambda__151,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_py(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_151
tff(fact_8331_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ky(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_152
tff(fact_8332_ATP_Olambda__153,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_ek(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_153
tff(fact_8333_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_acp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_8334_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_dz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_155
tff(fact_8335_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aa(list(A),fun(array(A),assn),aTP_Lamp_cs(list(A),fun(array(A),assn)),Uu),Uua) = aa(list(A),assn,snga_assn(A,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_8336_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_qf(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,dvd_dvd(nat,Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_8337_ATP_Olambda__158,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_qq(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,dvd_dvd(int,Uua),Uu) ) ).

% ATP.lambda_158
tff(fact_8338_ATP_Olambda__159,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_qh(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,dvd_dvd(A,Uua),Uu) ) ) ).

% ATP.lambda_159
tff(fact_8339_ATP_Olambda__160,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ee(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_160
tff(fact_8340_ATP_Olambda__161,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_lk(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu) ).

% ATP.lambda_161
tff(fact_8341_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mf(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_162
tff(fact_8342_ATP_Olambda__163,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aTP_Lamp_di(list(A),fun(A,list(A)),Uu),Uua) = aa(list(A),list(A),cons(A,Uua),Uu) ).

% ATP.lambda_163
tff(fact_8343_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(A,$o)),Uu),Uua)
    <=> aa(set(A),$o,member(A,Uua),Uu) ) ).

% ATP.lambda_164
tff(fact_8344_ATP_Olambda__165,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_xx(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_165
tff(fact_8345_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_kr(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_166
tff(fact_8346_ATP_Olambda__167,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ca(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_167
tff(fact_8347_ATP_Olambda__168,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_iw(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_168
tff(fact_8348_ATP_Olambda__169,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_iv(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_169
tff(fact_8349_ATP_Olambda__170,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ao(fun(A,nat),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_170
tff(fact_8350_ATP_Olambda__171,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_qn(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_8351_ATP_Olambda__172,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_qm(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_8352_ATP_Olambda__173,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_zy(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_173
tff(fact_8353_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_174
tff(fact_8354_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_sg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_175
tff(fact_8355_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_of(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_176
tff(fact_8356_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_177
tff(fact_8357_ATP_Olambda__178,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_lj(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_178
tff(fact_8358_ATP_Olambda__179,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_ey(B,fun(fun(B,A),A),Uu),Uua) = aa(B,A,Uua,Uu) ) ).

% ATP.lambda_179
tff(fact_8359_ATP_Olambda__180,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_abh(B,fun(fun(B,A),A),Uu),Uua) = aa(B,A,Uua,Uu) ) ).

% ATP.lambda_180
tff(fact_8360_ATP_Olambda__181,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_fj(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_181
tff(fact_8361_ATP_Olambda__182,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_abs(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_182
tff(fact_8362_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_wy(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).

% ATP.lambda_183
tff(fact_8363_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_xn(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_184
tff(fact_8364_ATP_Olambda__185,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_xi(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_185
tff(fact_8365_ATP_Olambda__186,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option($o)),aTP_Lamp_yq(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Uu),Uua) = aa(fun(A,fun(list(A),option($o))),fun(product_prod(A,list(A)),option($o)),product_case_prod(A,list(A),option($o)),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_yp(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua)) ).

% ATP.lambda_186
tff(fact_8366_ATP_Olambda__187,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_wt(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_ws(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_187
tff(fact_8367_ATP_Olambda__188,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),list(A)),aTP_Lamp_vu(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu),Uua) = aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_vt(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_188
tff(fact_8368_ATP_Olambda__189,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C] : aa(C,fun(product_prod(B,A),C),aTP_Lamp_up(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),Uu),Uua) = aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_uo(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).

% ATP.lambda_189
tff(fact_8369_ATP_Olambda__190,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,assn)),Uua: A] : aa(A,assn,aTP_Lamp_zt(fun(B,fun(A,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),aTP_Lamp_zs(fun(B,fun(A,assn)),fun(A,fun(B,assn)),Uu),Uua)) ).

% ATP.lambda_190
tff(fact_8370_ATP_Olambda__191,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_ack(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ach(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_191
tff(fact_8371_ATP_Olambda__192,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_acg(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_acf(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_192
tff(fact_8372_ATP_Olambda__193,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_aci(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ach(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_193
tff(fact_8373_ATP_Olambda__194,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_acj(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_acf(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_194
tff(fact_8374_ATP_Olambda__195,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aag(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_195
tff(fact_8375_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_ts(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_196
tff(fact_8376_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ua(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_197
tff(fact_8377_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tf(A,fun(nat,A),Uu),Uua) = uminus_uminus(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),uminus_uminus(A,Uu)),Uua))) ) ).

% ATP.lambda_198
tff(fact_8378_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_td(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_199
tff(fact_8379_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_te(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_200
tff(fact_8380_ATP_Olambda__201,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_fg(set(set(A)),fun(A,$o),Uu),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(set(A)),set($o),image(set(A),$o,member(A,Uua)),Uu)) ) ).

% ATP.lambda_201
tff(fact_8381_ATP_Olambda__202,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_acl(set(set(A)),fun(A,$o),Uu),Uua)
    <=> complete_Inf_Inf($o,aa(set(set(A)),set($o),image(set(A),$o,member(A,Uua)),Uu)) ) ).

% ATP.lambda_202
tff(fact_8382_ATP_Olambda__203,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_zj(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_203
tff(fact_8383_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_lr(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_204
tff(fact_8384_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dm(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_205
tff(fact_8385_ATP_Olambda__206,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_qc(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_206
tff(fact_8386_ATP_Olambda__207,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_fw(fun(B,set(A)),fun(set(B),set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),Uu),Uua)) ).

% ATP.lambda_207
tff(fact_8387_ATP_Olambda__208,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_ace(fun(B,set(A)),fun(set(B),set(A)),Uu),Uua) = complete_Inf_Inf(set(A),aa(set(B),set(set(A)),image(B,set(A),Uu),Uua)) ).

% ATP.lambda_208
tff(fact_8388_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_wu(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_209
tff(fact_8389_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_ww(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_210
tff(fact_8390_ATP_Olambda__211,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_em(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_211
tff(fact_8391_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu),Uua)) ).

% ATP.lambda_212
tff(fact_8392_ATP_Olambda__213,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yw(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu)) ).

% ATP.lambda_213
tff(fact_8393_ATP_Olambda__214,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_xu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).

% ATP.lambda_214
tff(fact_8394_ATP_Olambda__215,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_xv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).

% ATP.lambda_215
tff(fact_8395_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_mp(A,fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uua) ) ) ).

% ATP.lambda_216
tff(fact_8396_ATP_Olambda__217,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ei(set(A),fun(A,$o),Uu),Uua)
    <=> ~ aa(set(A),$o,member(A,Uua),Uu) ) ).

% ATP.lambda_217
tff(fact_8397_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_kz(A,fun(A,$o),Uu),Uua)
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_218
tff(fact_8398_ATP_Olambda__219,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_li(A,fun(A,$o),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_219
tff(fact_8399_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_rz(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_220
tff(fact_8400_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_rq(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_221
tff(fact_8401_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ri(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_222
tff(fact_8402_ATP_Olambda__223,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_jc(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_223
tff(fact_8403_ATP_Olambda__224,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_oa(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_224
tff(fact_8404_ATP_Olambda__225,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_nw(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_225
tff(fact_8405_ATP_Olambda__226,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_aak(fun(A,$o),fun(A,B),Uu),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uua)) ) ).

% ATP.lambda_226
tff(fact_8406_ATP_Olambda__227,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_no(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_227
tff(fact_8407_ATP_Olambda__228,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_gk(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_228
tff(fact_8408_ATP_Olambda__229,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_dc(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_229
tff(fact_8409_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_hu(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_230
tff(fact_8410_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pj(fun(A,real),fun(A,real),Uu),Uua) = ln_ln(real,aa(A,real,Uu,Uua)) ).

% ATP.lambda_231
tff(fact_8411_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rf(fun(nat,A),fun(nat,A),Uu),Uua) = uminus_uminus(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_232
tff(fact_8412_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tn(fun(nat,A),fun(nat,A),Uu),Uua) = uminus_uminus(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_233
tff(fact_8413_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tj(fun(nat,A),fun(nat,A),Uu),Uua) = uminus_uminus(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_234
tff(fact_8414_ATP_Olambda__235,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_abl(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = uminus_uminus(set(A),aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_235
tff(fact_8415_ATP_Olambda__236,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_abp(fun(B,A),fun(B,A),Uu),Uua) = uminus_uminus(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_8416_ATP_Olambda__237,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ha(fun(B,A),fun(B,A),Uu),Uua) = uminus_uminus(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_237
tff(fact_8417_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ov(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_238
tff(fact_8418_ATP_Olambda__239,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_pk(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_239
tff(fact_8419_ATP_Olambda__240,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_pl(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_240
tff(fact_8420_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_jo(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_241
tff(fact_8421_ATP_Olambda__242,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_rh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_242
tff(fact_8422_ATP_Olambda__243,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_gl(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_243
tff(fact_8423_ATP_Olambda__244,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_oc(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_244
tff(fact_8424_ATP_Olambda__245,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ol(fun(A,B),fun(A,B),Uu),Uua) = exp(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_245
tff(fact_8425_ATP_Olambda__246,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_acd(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_246
tff(fact_8426_ATP_Olambda__247,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_ep(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_8427_ATP_Olambda__248,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: A] : aa(A,assn,aTP_Lamp_zx(fun(A,fun(B,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),Uu,Uua)) ).

% ATP.lambda_248
tff(fact_8428_ATP_Olambda__249,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_uj(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).

% ATP.lambda_249
tff(fact_8429_ATP_Olambda__250,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_aaw(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_250
tff(fact_8430_ATP_Olambda__251,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_rp(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_8431_ATP_Olambda__252,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_ge(fun(B,list(A)),fun(B,set(A)),Uu),Uua) = aa(list(A),set(A),set2(A),aa(B,list(A),Uu,Uua)) ).

% ATP.lambda_252
tff(fact_8432_ATP_Olambda__253,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_ef(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = insert(B,aa(A,B,Uu,Uua)) ).

% ATP.lambda_253
tff(fact_8433_ATP_Olambda__254,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_pi(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_254
tff(fact_8434_ATP_Olambda__255,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_id(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_255
tff(fact_8435_ATP_Olambda__256,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ms(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_256
tff(fact_8436_ATP_Olambda__257,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_eh(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_257
tff(fact_8437_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_aai(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_258
tff(fact_8438_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_df(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).

% ATP.lambda_259
tff(fact_8439_ATP_Olambda__260,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_so(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_260
tff(fact_8440_ATP_Olambda__261,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ql(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_261
tff(fact_8441_ATP_Olambda__262,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_us(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_262
tff(fact_8442_ATP_Olambda__263,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_uq(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_263
tff(fact_8443_ATP_Olambda__264,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ur(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,minus_minus(int,Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_264
tff(fact_8444_ATP_Olambda__265,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_ut(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,minus_minus(A,Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_265
tff(fact_8445_ATP_Olambda__266,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_rc(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(set(nat),$o,member(nat,Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_266
tff(fact_8446_ATP_Olambda__267,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_ne(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_267
tff(fact_8447_ATP_Olambda__268,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_oj(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_268
tff(fact_8448_ATP_Olambda__269,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_gn(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_269
tff(fact_8449_ATP_Olambda__270,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_nq(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_270
tff(fact_8450_ATP_Olambda__271,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_qw(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_271
tff(fact_8451_ATP_Olambda__272,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_gm(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_272
tff(fact_8452_ATP_Olambda__273,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_np(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_273
tff(fact_8453_ATP_Olambda__274,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_la(fun(A,$o),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(list(A),list(A),cons(A,Uua),Uub),Uub) ).

% ATP.lambda_274
tff(fact_8454_ATP_Olambda__275,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_dn(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),insert(A,Uua),Uub),Uub) ).

% ATP.lambda_275
tff(fact_8455_ATP_Olambda__276,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_rb(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_276
tff(fact_8456_ATP_Olambda__277,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
          aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_lf(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_277
tff(fact_8457_ATP_Olambda__278,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,$o),fun(A,B),aTP_Lamp_hf(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_278
tff(fact_8458_ATP_Olambda__279,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,$o),fun(A,B),aTP_Lamp_od(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_279
tff(fact_8459_ATP_Olambda__280,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_mo(fun(B,A),fun(fun(B,$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_280
tff(fact_8460_ATP_Olambda__281,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: option(A),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_wv(option(A),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,Uu,aTP_Lamp_wu(A,fun(A,option(A)),Uub),Uua) ) ).

% ATP.lambda_281
tff(fact_8461_ATP_Olambda__282,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_vm(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).

% ATP.lambda_282
tff(fact_8462_ATP_Olambda__283,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: $o] :
      aa($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_zi(set(A),fun(set(A),fun($o,set(A))),Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_283
tff(fact_8463_ATP_Olambda__284,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ach(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ).

% ATP.lambda_284
tff(fact_8464_ATP_Olambda__285,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_vc(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_8465_ATP_Olambda__286,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_va(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_286
tff(fact_8466_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_or(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_8467_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_jm(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_8468_ATP_Olambda__289,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_acf(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_289
tff(fact_8469_ATP_Olambda__290,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_fa(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_290
tff(fact_8470_ATP_Olambda__291,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_abk(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_291
tff(fact_8471_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_fl(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_292
tff(fact_8472_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_ny(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_293
tff(fact_8473_ATP_Olambda__294,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_gr(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_294
tff(fact_8474_ATP_Olambda__295,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,assn)),Uua: A,Uub: B] : aa(B,assn,aa(A,fun(B,assn),aTP_Lamp_zs(fun(B,fun(A,assn)),fun(A,fun(B,assn)),Uu),Uua),Uub) = aa(A,assn,aa(B,fun(A,assn),Uu,Uub),Uua) ).

% ATP.lambda_295
tff(fact_8475_ATP_Olambda__296,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ko(fun(B,fun(A,A)),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(B,fun(A,A),Uu,Uub),Uua) ).

% ATP.lambda_296
tff(fact_8476_ATP_Olambda__297,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_rl(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(A,fun(nat,B),Uu,Uub),Uua) ) ).

% ATP.lambda_297
tff(fact_8477_ATP_Olambda__298,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,assn)),Uua: B,Uub: A] : aa(A,assn,aa(B,fun(A,assn),aTP_Lamp_zw(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Uu),Uua),Uub) = aa(B,assn,aa(A,fun(B,assn),Uu,Uub),Uua) ).

% ATP.lambda_298
tff(fact_8478_ATP_Olambda__299,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_nu(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_299
tff(fact_8479_ATP_Olambda__300,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_hd(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_300
tff(fact_8480_ATP_Olambda__301,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,A)),Uua: B,Uub: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_kp(fun(A,fun(B,A)),fun(B,fun(A,A)),Uu),Uua),Uub) = aa(B,A,aa(A,fun(B,A),Uu,Uub),Uua) ).

% ATP.lambda_301
tff(fact_8481_ATP_Olambda__302,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_kn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_302
tff(fact_8482_ATP_Olambda__303,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_sw(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_sv(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_303
tff(fact_8483_ATP_Olambda__304,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_su(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_st(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_304
tff(fact_8484_ATP_Olambda__305,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_ss(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_sr(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_305
tff(fact_8485_ATP_Olambda__306,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_rs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_rr(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_306
tff(fact_8486_ATP_Olambda__307,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_si(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_307
tff(fact_8487_ATP_Olambda__308,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ys(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_308
tff(fact_8488_ATP_Olambda__309,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_os(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_or(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_309
tff(fact_8489_ATP_Olambda__310,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_jn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_310
tff(fact_8490_ATP_Olambda__311,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ym(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yl(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_311
tff(fact_8491_ATP_Olambda__312,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_yk(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yj(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_312
tff(fact_8492_ATP_Olambda__313,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_yi(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yh(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_313
tff(fact_8493_ATP_Olambda__314,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_yg(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yf(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_314
tff(fact_8494_ATP_Olambda__315,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ye(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_yd(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_315
tff(fact_8495_ATP_Olambda__316,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_yb(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ya(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_316
tff(fact_8496_ATP_Olambda__317,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_nz(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(C,fun(B,A),aTP_Lamp_ny(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_317
tff(fact_8497_ATP_Olambda__318,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(nat,B)),Uub: nat] : aa(nat,B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_rm(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),Uu),Uua),Uub) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(nat,fun(A,B),aTP_Lamp_rl(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_318
tff(fact_8498_ATP_Olambda__319,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_gs(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(C,fun(B,A),aTP_Lamp_gr(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_319
tff(fact_8499_ATP_Olambda__320,axiom,
    ! [D6: $tType,E4: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: product_prod(B,C),Uub: product_prod(D6,E4)] : aa(product_prod(D6,E4),set(A),aa(product_prod(B,C),fun(product_prod(D6,E4),set(A)),aTP_Lamp_uw(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(B,C),fun(product_prod(D6,E4),set(A))),Uu),Uua),Uub) = aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(product_prod(D6,E4),fun(B,fun(C,set(A))),aTP_Lamp_uv(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(D6,E4),fun(B,fun(C,set(A)))),Uu),Uub)),Uua) ).

% ATP.lambda_320
tff(fact_8500_ATP_Olambda__321,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_lo(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_321
tff(fact_8501_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_im(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_322
tff(fact_8502_ATP_Olambda__323,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_iy(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_323
tff(fact_8503_ATP_Olambda__324,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_lp(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_324
tff(fact_8504_ATP_Olambda__325,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_abb(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,minus_minus(nat,Uu),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_325
tff(fact_8505_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_ti(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_326
tff(fact_8506_ATP_Olambda__327,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_abc(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_327
tff(fact_8507_ATP_Olambda__328,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_aav(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_328
tff(fact_8508_ATP_Olambda__329,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_co(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_329
tff(fact_8509_ATP_Olambda__330,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_bk(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_330
tff(fact_8510_ATP_Olambda__331,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_bi(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_331
tff(fact_8511_ATP_Olambda__332,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_bj(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_332
tff(fact_8512_ATP_Olambda__333,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_aba(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_333
tff(fact_8513_ATP_Olambda__334,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aap(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_334
tff(fact_8514_ATP_Olambda__335,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_nm(set(list(A)),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(list(A)),$o,member(list(A),append(A,Uua,aa(list(A),list(A),cons(A,Uub),nil(A)))),Uu) ) ).

% ATP.lambda_335
tff(fact_8515_ATP_Olambda__336,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_aan(nat,fun(nat,fun(set(nat),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(nat)),$o,member(set(nat),Uub),pow(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_336
tff(fact_8516_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_aat(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$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_337
tff(fact_8517_ATP_Olambda__338,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_kw(fun(A,$o),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_338
tff(fact_8518_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pc(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_339
tff(fact_8519_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_aao(set(A),fun(nat,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu)
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_340
tff(fact_8520_ATP_Olambda__341,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aaq(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_341
tff(fact_8521_ATP_Olambda__342,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_th(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),diffs(A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_342
tff(fact_8522_ATP_Olambda__343,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_tl(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),diffs(A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_343
tff(fact_8523_ATP_Olambda__344,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_tm(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),diffs(A,Uua,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_344
tff(fact_8524_ATP_Olambda__345,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_mi(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_345
tff(fact_8525_ATP_Olambda__346,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ds(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_346
tff(fact_8526_ATP_Olambda__347,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ce(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_347
tff(fact_8527_ATP_Olambda__348,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_wf(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_348
tff(fact_8528_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A),Uua: word(A),Uub: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),aTP_Lamp_ch(word(A),fun(word(A),fun(word(A),$o)),Uu),Uua),Uub)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uu),Uub)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uub),Uua) ) ) ) ).

% ATP.lambda_349
tff(fact_8529_ATP_Olambda__350,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_do(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_350
tff(fact_8530_ATP_Olambda__351,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_wg(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_351
tff(fact_8531_ATP_Olambda__352,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dq(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_352
tff(fact_8532_ATP_Olambda__353,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dr(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_353
tff(fact_8533_ATP_Olambda__354,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dp(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_354
tff(fact_8534_ATP_Olambda__355,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ai(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_355
tff(fact_8535_ATP_Olambda__356,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_al(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_356
tff(fact_8536_ATP_Olambda__357,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_mt(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_357
tff(fact_8537_ATP_Olambda__358,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_by(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),set(A),minus_minus(set(A),Uub),Uu) = aa(set(A),set(A),minus_minus(set(A),Uua),Uu) ) ) ).

% ATP.lambda_358
tff(fact_8538_ATP_Olambda__359,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ic(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_359
tff(fact_8539_ATP_Olambda__360,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_ia(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_360
tff(fact_8540_ATP_Olambda__361,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_361
tff(fact_8541_ATP_Olambda__362,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_dt(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_362
tff(fact_8542_ATP_Olambda__363,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_363
tff(fact_8543_ATP_Olambda__364,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_uy(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_364
tff(fact_8544_ATP_Olambda__365,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cf(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_365
tff(fact_8545_ATP_Olambda__366,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,$o),aTP_Lamp_bg(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_366
tff(fact_8546_ATP_Olambda__367,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,$o),aTP_Lamp_ci(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_367
tff(fact_8547_ATP_Olambda__368,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,$o),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_368
tff(fact_8548_ATP_Olambda__369,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,$o),aTP_Lamp_cj(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_369
tff(fact_8549_ATP_Olambda__370,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ck(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_370
tff(fact_8550_ATP_Olambda__371,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,$o),aTP_Lamp_abd(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ~ aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_371
tff(fact_8551_ATP_Olambda__372,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa($o,assn,aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_bf(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Uu),Uua),(Uub)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uub) = vEBT_VEBT_minNull(Uu))) ).

% ATP.lambda_372
tff(fact_8552_ATP_Olambda__373,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ak(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_mint(Uu))) ).

% ATP.lambda_373
tff(fact_8553_ATP_Olambda__374,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ah(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_maxt(Uu))) ).

% ATP.lambda_374
tff(fact_8554_ATP_Olambda__375,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_cx(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_375
tff(fact_8555_ATP_Olambda__376,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_cu(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_376
tff(fact_8556_ATP_Olambda__377,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_cc(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & ~ aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_377
tff(fact_8557_ATP_Olambda__378,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jg(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_378
tff(fact_8558_ATP_Olambda__379,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_uz(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_379
tff(fact_8559_ATP_Olambda__380,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ve(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_380
tff(fact_8560_ATP_Olambda__381,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_jt(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_381
tff(fact_8561_ATP_Olambda__382,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_jr(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_382
tff(fact_8562_ATP_Olambda__383,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_js(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_383
tff(fact_8563_ATP_Olambda__384,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_jq(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_384
tff(fact_8564_ATP_Olambda__385,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ba(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu) ) ).

% ATP.lambda_385
tff(fact_8565_ATP_Olambda__386,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kt(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_386
tff(fact_8566_ATP_Olambda__387,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_gg(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_387
tff(fact_8567_ATP_Olambda__388,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_aau(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_388
tff(fact_8568_ATP_Olambda__389,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_wb(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_389
tff(fact_8569_ATP_Olambda__390,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_rx(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_390
tff(fact_8570_ATP_Olambda__391,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_rw(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_391
tff(fact_8571_ATP_Olambda__392,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sa(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_392
tff(fact_8572_ATP_Olambda__393,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_iq(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_393
tff(fact_8573_ATP_Olambda__394,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_ik(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_394
tff(fact_8574_ATP_Olambda__395,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ix(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_395
tff(fact_8575_ATP_Olambda__396,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_rv(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_396
tff(fact_8576_ATP_Olambda__397,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_tk(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_397
tff(fact_8577_ATP_Olambda__398,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_il(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_398
tff(fact_8578_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_in(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_399
tff(fact_8579_ATP_Olambda__400,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ay(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_400
tff(fact_8580_ATP_Olambda__401,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_vz(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% ATP.lambda_401
tff(fact_8581_ATP_Olambda__402,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_wa(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% ATP.lambda_402
tff(fact_8582_ATP_Olambda__403,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_bq(fun(B,A),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_403
tff(fact_8583_ATP_Olambda__404,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_nr(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_404
tff(fact_8584_ATP_Olambda__405,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_ht(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_405
tff(fact_8585_ATP_Olambda__406,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_hs(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_406
tff(fact_8586_ATP_Olambda__407,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_ob(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_407
tff(fact_8587_ATP_Olambda__408,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cm(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_408
tff(fact_8588_ATP_Olambda__409,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_hr(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_409
tff(fact_8589_ATP_Olambda__410,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_410
tff(fact_8590_ATP_Olambda__411,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gz(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_411
tff(fact_8591_ATP_Olambda__412,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_sj(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_412
tff(fact_8592_ATP_Olambda__413,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_kd(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_413
tff(fact_8593_ATP_Olambda__414,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hv(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_414
tff(fact_8594_ATP_Olambda__415,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fv(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_415
tff(fact_8595_ATP_Olambda__416,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fp(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_416
tff(fact_8596_ATP_Olambda__417,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zv(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_417
tff(fact_8597_ATP_Olambda__418,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_abq(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_418
tff(fact_8598_ATP_Olambda__419,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_abr(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_419
tff(fact_8599_ATP_Olambda__420,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_re(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_420
tff(fact_8600_ATP_Olambda__421,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_gu(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_421
tff(fact_8601_ATP_Olambda__422,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_cl(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_422
tff(fact_8602_ATP_Olambda__423,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [Uu: fun(B,word(A)),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_za(fun(B,word(A)),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(B,word(A),Uu,Uua)),aa(B,word(A),Uu,Uub)) ) ) ).

% ATP.lambda_423
tff(fact_8603_ATP_Olambda__424,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_ej(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_424
tff(fact_8604_ATP_Olambda__425,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_av(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_425
tff(fact_8605_ATP_Olambda__426,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aaa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_426
tff(fact_8606_ATP_Olambda__427,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_au(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_427
tff(fact_8607_ATP_Olambda__428,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_kv(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_428
tff(fact_8608_ATP_Olambda__429,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acq(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_429
tff(fact_8609_ATP_Olambda__430,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Uu: A,Uua: fun(A,word(B)),Uub: A] :
          ( aa(A,$o,aa(fun(A,word(B)),fun(A,$o),aTP_Lamp_aac(A,fun(fun(A,word(B)),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,word(B),Uua,Uu) = aa(A,word(B),Uua,Uub) ) ) ) ).

% ATP.lambda_430
tff(fact_8610_ATP_Olambda__431,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_nh(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_431
tff(fact_8611_ATP_Olambda__432,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ty(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_432
tff(fact_8612_ATP_Olambda__433,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zu(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),ex_assn(A,Uua)) ).

% ATP.lambda_433
tff(fact_8613_ATP_Olambda__434,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ta(fun(nat,real),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_434
tff(fact_8614_ATP_Olambda__435,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(B,real),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_sz(fun(B,real),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_435
tff(fact_8615_ATP_Olambda__436,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_nx(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_436
tff(fact_8616_ATP_Olambda__437,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_gq(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_437
tff(fact_8617_ATP_Olambda__438,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ax(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_438
tff(fact_8618_ATP_Olambda__439,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_gb(fun(B,set(A)),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua) ) ).

% ATP.lambda_439
tff(fact_8619_ATP_Olambda__440,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_kb(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_440
tff(fact_8620_ATP_Olambda__441,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_ra(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_441
tff(fact_8621_ATP_Olambda__442,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_iz(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_442
tff(fact_8622_ATP_Olambda__443,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qi(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_443
tff(fact_8623_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,$o),aTP_Lamp_ni(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_444
tff(fact_8624_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_rj(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_8625_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_gx(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_8626_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_aa(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_447
tff(fact_8627_ATP_Olambda__448,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_sf(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_448
tff(fact_8628_ATP_Olambda__449,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ke(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_449
tff(fact_8629_ATP_Olambda__450,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_fu(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),minus_minus(set(A),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_450
tff(fact_8630_ATP_Olambda__451,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_ns(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_451
tff(fact_8631_ATP_Olambda__452,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_et(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_452
tff(fact_8632_ATP_Olambda__453,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_abz(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_453
tff(fact_8633_ATP_Olambda__454,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_zo(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),Uua) ).

% ATP.lambda_454
tff(fact_8634_ATP_Olambda__455,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_nc(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_455
tff(fact_8635_ATP_Olambda__456,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_mu(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_456
tff(fact_8636_ATP_Olambda__457,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_zr(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(A,assn,Uu,Uub)),Uua) ).

% ATP.lambda_457
tff(fact_8637_ATP_Olambda__458,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_abn(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_458
tff(fact_8638_ATP_Olambda__459,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_er(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_459
tff(fact_8639_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_acn(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_460
tff(fact_8640_ATP_Olambda__461,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Uu: fun(A,word(B)),Uua: word(B),Uub: A] :
          ( aa(A,$o,aa(word(B),fun(A,$o),aTP_Lamp_md(fun(A,word(B)),fun(word(B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,word(B),Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_461
tff(fact_8641_ATP_Olambda__462,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_nj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_462
tff(fact_8642_ATP_Olambda__463,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cd(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_463
tff(fact_8643_ATP_Olambda__464,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pb(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu),Uub))) ) ).

% ATP.lambda_464
tff(fact_8644_ATP_Olambda__465,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_sb(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_465
tff(fact_8645_ATP_Olambda__466,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tz(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_466
tff(fact_8646_ATP_Olambda__467,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_tv(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_467
tff(fact_8647_ATP_Olambda__468,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] : aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_vx(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua),Uub) = case_list(list(A),A,Uua,aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_vw(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub)) ) ).

% ATP.lambda_468
tff(fact_8648_ATP_Olambda__469,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_yt(nat,fun(nat,fun(word(A),$o)),Uu),Uua),Uub)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Uub),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uu)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uua))) ) ) ).

% ATP.lambda_469
tff(fact_8649_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_yu(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_470
tff(fact_8650_ATP_Olambda__471,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_ot(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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_471
tff(fact_8651_ATP_Olambda__472,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_sl(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(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_472
tff(fact_8652_ATP_Olambda__473,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_kf(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(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_473
tff(fact_8653_ATP_Olambda__474,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: list(A),Uub: B] : aa(B,list(A),aa(list(A),fun(B,list(A)),aTP_Lamp_gt(fun(B,A),fun(list(A),fun(B,list(A))),Uu),Uua),Uub) = append(A,Uua,aa(list(A),list(A),cons(A,aa(B,A,Uu,Uub)),nil(A))) ).

% ATP.lambda_474
tff(fact_8654_ATP_Olambda__475,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_pt(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_475
tff(fact_8655_ATP_Olambda__476,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_pp(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_476
tff(fact_8656_ATP_Olambda__477,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_lq(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),map(nat,A,Uua,upt(zero_zero(nat),Uu))) ) ).

% ATP.lambda_477
tff(fact_8657_ATP_Olambda__478,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_kl(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_478
tff(fact_8658_ATP_Olambda__479,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_hn(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_479
tff(fact_8659_ATP_Olambda__480,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_pr(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_480
tff(fact_8660_ATP_Olambda__481,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_dd(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),replicate(A,Uu,Uua)) ) ).

% ATP.lambda_481
tff(fact_8661_ATP_Olambda__482,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_vf(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_482
tff(fact_8662_ATP_Olambda__483,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_ud(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uub)) ).

% ATP.lambda_483
tff(fact_8663_ATP_Olambda__484,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_xy(A,fun(list(A),fun(nat,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),cons(A,Uu),take(A,Uub,Uua)) ).

% ATP.lambda_484
tff(fact_8664_ATP_Olambda__485,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_xd(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_485
tff(fact_8665_ATP_Olambda__486,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: real,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sy(real,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_486
tff(fact_8666_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_sx(fun(nat,A),fun(real,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_487
tff(fact_8667_ATP_Olambda__488,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ox(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_488
tff(fact_8668_ATP_Olambda__489,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ng(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_489
tff(fact_8669_ATP_Olambda__490,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_aab(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_490
tff(fact_8670_ATP_Olambda__491,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_mq(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_491
tff(fact_8671_ATP_Olambda__492,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_nk(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_492
tff(fact_8672_ATP_Olambda__493,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zn(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_493
tff(fact_8673_ATP_Olambda__494,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_qz(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_494
tff(fact_8674_ATP_Olambda__495,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_gy(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_495
tff(fact_8675_ATP_Olambda__496,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_se(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_496
tff(fact_8676_ATP_Olambda__497,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_rk(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_497
tff(fact_8677_ATP_Olambda__498,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_ab(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_498
tff(fact_8678_ATP_Olambda__499,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_abm(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),minus_minus(set(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_499
tff(fact_8679_ATP_Olambda__500,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_og(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_500
tff(fact_8680_ATP_Olambda__501,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_zp(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Uu),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_501
tff(fact_8681_ATP_Olambda__502,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_eu(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_502
tff(fact_8682_ATP_Olambda__503,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_abx(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_503
tff(fact_8683_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_zq(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Uu),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_504
tff(fact_8684_ATP_Olambda__505,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_nb(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_505
tff(fact_8685_ATP_Olambda__506,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abo(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_506
tff(fact_8686_ATP_Olambda__507,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mw(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_507
tff(fact_8687_ATP_Olambda__508,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_qj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,dvd_dvd(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_508
tff(fact_8688_ATP_Olambda__509,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_uk(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).

% ATP.lambda_509
tff(fact_8689_ATP_Olambda__510,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [Uu: fun(B,word(A)),Uua: word(A),Uub: B] :
          ( aa(B,$o,aa(word(A),fun(B,$o),aTP_Lamp_zb(fun(B,word(A)),fun(word(A),fun(B,$o)),Uu),Uua),Uub)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Uua),aa(B,word(A),Uu,Uub)) ) ) ).

% ATP.lambda_510
tff(fact_8690_ATP_Olambda__511,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_fh(fun(B,set(A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,Uua)) ) ).

% ATP.lambda_511
tff(fact_8691_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ft(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_512
tff(fact_8692_ATP_Olambda__513,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ev(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),insert(A,Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_513
tff(fact_8693_ATP_Olambda__514,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_fr(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_514
tff(fact_8694_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_acv(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_515
tff(fact_8695_ATP_Olambda__516,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: fun(list(word(A)),word(A)),Uua: list(word(A)),Uub: word(A)] :
          ( aa(word(A),$o,aa(list(word(A)),fun(word(A),$o),aTP_Lamp_yy(fun(list(word(A)),word(A)),fun(list(word(A)),fun(word(A),$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(word(A)),word(A),Uu,Uua) ) ) ) ).

% ATP.lambda_516
tff(fact_8696_ATP_Olambda__517,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_lb(fun(list(A),A),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_517
tff(fact_8697_ATP_Olambda__518,axiom,
    ! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_qv(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_518
tff(fact_8698_ATP_Olambda__519,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pa(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_519
tff(fact_8699_ATP_Olambda__520,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_lt(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = append(A,Uub,Uu) ).

% ATP.lambda_520
tff(fact_8700_ATP_Olambda__521,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aq(fun(A,nat),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$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_521
tff(fact_8701_ATP_Olambda__522,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,nat),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat))) ) ).

% ATP.lambda_522
tff(fact_8702_ATP_Olambda__523,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_bt(fun(A,nat),fun(fun(A,nat),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$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_523
tff(fact_8703_ATP_Olambda__524,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_om(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_524
tff(fact_8704_ATP_Olambda__525,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_kc(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_525
tff(fact_8705_ATP_Olambda__526,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_jk(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_526
tff(fact_8706_ATP_Olambda__527,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qy(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_527
tff(fact_8707_ATP_Olambda__528,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_oh(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_528
tff(fact_8708_ATP_Olambda__529,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_hj(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_529
tff(fact_8709_ATP_Olambda__530,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_vo(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_530
tff(fact_8710_ATP_Olambda__531,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_lv(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_531
tff(fact_8711_ATP_Olambda__532,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_sh(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_532
tff(fact_8712_ATP_Olambda__533,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,C),Uub: B] : aa(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_fq(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),Uu),Uua),Uub) = aa(C,set(A),Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_533
tff(fact_8713_ATP_Olambda__534,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_zd(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_534
tff(fact_8714_ATP_Olambda__535,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_xt(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_535
tff(fact_8715_ATP_Olambda__536,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_xk(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_536
tff(fact_8716_ATP_Olambda__537,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ea(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_537
tff(fact_8717_ATP_Olambda__538,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_ze(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_538
tff(fact_8718_ATP_Olambda__539,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_aax(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_539
tff(fact_8719_ATP_Olambda__540,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C,Uub: B] : aa(B,fun(A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_uo(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).

% ATP.lambda_540
tff(fact_8720_ATP_Olambda__541,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_acr(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_541
tff(fact_8721_ATP_Olambda__542,axiom,
    ! [B: $tType,C: $tType,A: $tType,E4: $tType,D6: $tType,Uu: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: set(product_prod(D6,E4)),Uub: product_prod(B,C)] : aa(product_prod(B,C),set(A),aa(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A)),aTP_Lamp_ux(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(product_prod(B,C),set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D6,E4)),set(set(A)),image(product_prod(D6,E4),set(A),aa(product_prod(B,C),fun(product_prod(D6,E4),set(A)),aTP_Lamp_uw(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(B,C),fun(product_prod(D6,E4),set(A))),Uu),Uub)),Uua)) ).

% ATP.lambda_542
tff(fact_8722_ATP_Olambda__543,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_fm(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(C,fun(B,A),aTP_Lamp_fl(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_543
tff(fact_8723_ATP_Olambda__544,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_abu(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = complete_Inf_Inf(A,aa(set(B),set(A),image(B,A,aa(C,fun(B,A),aTP_Lamp_fl(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_544
tff(fact_8724_ATP_Olambda__545,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_rd(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_545
tff(fact_8725_ATP_Olambda__546,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_oy(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_546
tff(fact_8726_ATP_Olambda__547,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_fk(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_547
tff(fact_8727_ATP_Olambda__548,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_abt(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_548
tff(fact_8728_ATP_Olambda__549,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_cz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_549
tff(fact_8729_ATP_Olambda__550,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_db(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_550
tff(fact_8730_ATP_Olambda__551,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_qa(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_high(Uu,Uua)) ).

% ATP.lambda_551
tff(fact_8731_ATP_Olambda__552,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_pz(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_low(Uu,Uua)) ).

% ATP.lambda_552
tff(fact_8732_ATP_Olambda__553,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_fn(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),Uu),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_553
tff(fact_8733_ATP_Olambda__554,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_fo(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),Uu),aa(C,set(B),Uua,Uub))) ).

% ATP.lambda_554
tff(fact_8734_ATP_Olambda__555,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_fy(fun(B,A),fun(fun(C,set(B)),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub))) ) ).

% ATP.lambda_555
tff(fact_8735_ATP_Olambda__556,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_aca(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu),Uua),Uub) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),Uu),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_556
tff(fact_8736_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_qk(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ aa(B,$o,dvd_dvd(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_557
tff(fact_8737_ATP_Olambda__558,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_aal(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_hb(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_558
tff(fact_8738_ATP_Olambda__559,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,Uu: fun(B,fun(C,fun(D6,A))),Uua: D6,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D6,fun(B,fun(C,A)),aTP_Lamp_uf(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D6,A,aa(C,fun(D6,A),aa(B,fun(C,fun(D6,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_559
tff(fact_8739_ATP_Olambda__560,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_sv(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_560
tff(fact_8740_ATP_Olambda__561,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_sr(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,uminus_uminus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_561
tff(fact_8741_ATP_Olambda__562,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_st(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),uminus_uminus(int,one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_562
tff(fact_8742_ATP_Olambda__563,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_it(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(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_563
tff(fact_8743_ATP_Olambda__564,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_pe(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(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_564
tff(fact_8744_ATP_Olambda__565,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_pf(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_565
tff(fact_8745_ATP_Olambda__566,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_iu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_566
tff(fact_8746_ATP_Olambda__567,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_acu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(set(A),$o,member(A,Uuc),Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_567
tff(fact_8747_ATP_Olambda__568,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_sk(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(set(nat),$o,member(nat,Uuc),Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_568
tff(fact_8748_ATP_Olambda__569,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_ow(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_569
tff(fact_8749_ATP_Olambda__570,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_hq(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_570
tff(fact_8750_ATP_Olambda__571,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_aaz(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_571
tff(fact_8751_ATP_Olambda__572,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_cb(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_572
tff(fact_8752_ATP_Olambda__573,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
      aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mr(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_573
tff(fact_8753_ATP_Olambda__574,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,$o),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_oo(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_574
tff(fact_8754_ATP_Olambda__575,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,$o),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_nf(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_575
tff(fact_8755_ATP_Olambda__576,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_bs(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ).

% ATP.lambda_576
tff(fact_8756_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType,C: $tType,D6: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D6,B),Uub: fun(D6,C),Uuc: D6] : aa(D6,A,aa(fun(D6,C),fun(D6,A),aa(fun(D6,B),fun(fun(D6,C),fun(D6,A)),aTP_Lamp_ug(fun(B,fun(C,A)),fun(fun(D6,B),fun(fun(D6,C),fun(D6,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D6,B,Uua,Uuc)),aa(D6,C,Uub,Uuc)) ).

% ATP.lambda_577
tff(fact_8757_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,A)),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,A,aa(A,fun(C,A),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_dh(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu,Uub),aa(C,B,Uua,Uuc)) ).

% ATP.lambda_578
tff(fact_8758_ATP_Olambda__579,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_pw(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_pv(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),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_579
tff(fact_8759_ATP_Olambda__580,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_gw(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gv(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_580
tff(fact_8760_ATP_Olambda__581,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A,Uuc: A] :
          aa(A,fun(list(A),list(A)),aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_vw(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub),Uuc) = case_list(list(A),A,
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uu,Uuc)),Uua,aa(list(A),list(A),cons(A,Uuc),aa(list(A),list(A),cons(A,Uub),nil(A)))),
            aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_vv(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_581
tff(fact_8761_ATP_Olambda__582,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_ig(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),
                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_582
tff(fact_8762_ATP_Olambda__583,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: $o,Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa($o,fun(A,fun(B,assn)),aTP_Lamp_ae(fun(A,fun(B,assn)),fun($o,fun(A,fun(B,assn))),Uu),(Uua)),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Uu,Uub),Uuc)),pure_assn((Uua))) ).

% ATP.lambda_583
tff(fact_8763_ATP_Olambda__584,axiom,
    ! [C: $tType,A: $tType,B: $tType,E4: $tType,D6: $tType,Uu: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: product_prod(D6,E4),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(product_prod(D6,E4),fun(B,fun(C,set(A))),aTP_Lamp_uv(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(product_prod(D6,E4),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(D6,E4),set(A),aa(fun(D6,fun(E4,set(A))),fun(product_prod(D6,E4),set(A)),product_case_prod(D6,E4,set(A)),aa(C,fun(D6,fun(E4,set(A))),aa(B,fun(C,fun(D6,fun(E4,set(A)))),Uu,Uub),Uuc)),Uua) ).

% ATP.lambda_584
tff(fact_8764_ATP_Olambda__585,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_nv(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_nu(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_aw(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_585
tff(fact_8765_ATP_Olambda__586,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_he(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_hd(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_aw(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_586
tff(fact_8766_ATP_Olambda__587,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_is(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,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_ir(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_587
tff(fact_8767_ATP_Olambda__588,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_ip(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_io(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_588
tff(fact_8768_ATP_Olambda__589,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_kh(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_589
tff(fact_8769_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_pv(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_590
tff(fact_8770_ATP_Olambda__591,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_vh(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
        & aa(product_prod(A,B),$o,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)) ) ) ).

% ATP.lambda_591
tff(fact_8771_ATP_Olambda__592,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_mm(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_592
tff(fact_8772_ATP_Olambda__593,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_mk(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_593
tff(fact_8773_ATP_Olambda__594,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_ml(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_594
tff(fact_8774_ATP_Olambda__595,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_tt(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_595
tff(fact_8775_ATP_Olambda__596,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_kj(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_596
tff(fact_8776_ATP_Olambda__597,axiom,
    ! [A: $tType,Uu: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_wo($o,fun(A,fun(A,fun(A,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_597
tff(fact_8777_ATP_Olambda__598,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_eb(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),aa(set(B),set(A),image(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_598
tff(fact_8778_ATP_Olambda__599,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_hb(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_599
tff(fact_8779_ATP_Olambda__600,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_aw(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_600
tff(fact_8780_ATP_Olambda__601,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_ki(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_601
tff(fact_8781_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_ed(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_602
tff(fact_8782_ATP_Olambda__603,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_hg(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_603
tff(fact_8783_ATP_Olambda__604,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_hk(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_604
tff(fact_8784_ATP_Olambda__605,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_aco(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uub)
        & aa(set(A),$o,member(A,aa(B,A,Uua,Uuc)),Uu) ) ) ).

% ATP.lambda_605
tff(fact_8785_ATP_Olambda__606,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_kk(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_606
tff(fact_8786_ATP_Olambda__607,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_mj(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_607
tff(fact_8787_ATP_Olambda__608,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: list(A),Uuc: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_vt(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua),Uub),Uuc) = append(A,aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uua),append(A,Uub,aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uuc))) ) ).

% ATP.lambda_608
tff(fact_8788_ATP_Olambda__609,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_yd(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_609
tff(fact_8789_ATP_Olambda__610,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ya(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_610
tff(fact_8790_ATP_Olambda__611,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_jd(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_611
tff(fact_8791_ATP_Olambda__612,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_je(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_612
tff(fact_8792_ATP_Olambda__613,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_wn(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_613
tff(fact_8793_ATP_Olambda__614,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ec(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_614
tff(fact_8794_ATP_Olambda__615,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_fi(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu,Uua)) ) ).

% ATP.lambda_615
tff(fact_8795_ATP_Olambda__616,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_bc(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,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_616
tff(fact_8796_ATP_Olambda__617,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_bh(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,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_617
tff(fact_8797_ATP_Olambda__618,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa($o,assn,aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_ag(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Uu),Uua),Uub),(Uuc)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uuc) = aa(nat,$o,vEBT_vebt_member(Uu),Uub))) ).

% ATP.lambda_618
tff(fact_8798_ATP_Olambda__619,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_ct(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_619
tff(fact_8799_ATP_Olambda__620,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_ry(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_620
tff(fact_8800_ATP_Olambda__621,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_gi(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_621
tff(fact_8801_ATP_Olambda__622,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_jw(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_622
tff(fact_8802_ATP_Olambda__623,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_cy(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_623
tff(fact_8803_ATP_Olambda__624,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_ir(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_624
tff(fact_8804_ATP_Olambda__625,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_rr(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_625
tff(fact_8805_ATP_Olambda__626,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_io(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_626
tff(fact_8806_ATP_Olambda__627,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_gv(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_627
tff(fact_8807_ATP_Olambda__628,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_acb(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_628
tff(fact_8808_ATP_Olambda__629,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(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_abv(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_629
tff(fact_8809_ATP_Olambda__630,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_mz(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_630
tff(fact_8810_ATP_Olambda__631,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(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_mx(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_631
tff(fact_8811_ATP_Olambda__632,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,Uu: fun(C,A),Uua: fun(D6,B),Uub: C,Uuc: D6] : aa(D6,product_prod(A,B),aa(C,fun(D6,product_prod(A,B)),aa(fun(D6,B),fun(C,fun(D6,product_prod(A,B))),aTP_Lamp_ul(fun(C,A),fun(fun(D6,B),fun(C,fun(D6,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D6,B,Uua,Uuc)) ).

% ATP.lambda_632
tff(fact_8812_ATP_Olambda__633,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [Uu: fun(B,word(A)),Uua: fun(list(B),word(A)),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),word(A)),fun(list(B),fun(B,$o)),aTP_Lamp_yz(fun(B,word(A)),fun(fun(list(B),word(A)),fun(list(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(B,word(A),Uu,Uuc) = aa(list(B),word(A),Uua,Uub) ) ) ) ).

% ATP.lambda_633
tff(fact_8813_ATP_Olambda__634,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_ld(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_634
tff(fact_8814_ATP_Olambda__635,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_nt(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_hb(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_635
tff(fact_8815_ATP_Olambda__636,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hc(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_hb(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_636
tff(fact_8816_ATP_Olambda__637,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_vv(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(list(A),list(A))),list(A),aa(fun(list(A),fun(product_prod(list(A),list(A)),list(A))),fun(product_prod(list(A),product_prod(list(A),list(A))),list(A)),product_case_prod(list(A),product_prod(list(A),list(A)),list(A)),aTP_Lamp_vu(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu)),linorder_part(A,B,Uu,aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Uua)) ) ).

% ATP.lambda_637
tff(fact_8817_ATP_Olambda__638,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_bn(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_638
tff(fact_8818_ATP_Olambda__639,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_bm(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_639
tff(fact_8819_ATP_Olambda__640,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_aay(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_hk(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_640
tff(fact_8820_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,assn)),Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_af($o,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),(Uu)),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((Uu))),aa(B,assn,aa(A,fun(B,assn),Uua,Uub),Uuc)) ).

% ATP.lambda_641
tff(fact_8821_ATP_Olambda__642,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_xw(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_642
tff(fact_8822_ATP_Olambda__643,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_bv(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_643
tff(fact_8823_ATP_Olambda__644,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_vr($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_644
tff(fact_8824_ATP_Olambda__645,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option($o),aa(A,fun(list(A),option($o)),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_yp(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua),Uub),Uuc) = subset_eq_mset_impl(A,Uu,append(A,Uua,Uuc)) ).

% ATP.lambda_645
tff(fact_8825_ATP_Olambda__646,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_oe(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_hg(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_646
tff(fact_8826_ATP_Olambda__647,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_oi(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_hk(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_647
tff(fact_8827_ATP_Olambda__648,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_hh(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_hg(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_648
tff(fact_8828_ATP_Olambda__649,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_hl(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_hk(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_649
tff(fact_8829_ATP_Olambda__650,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun($o,fun($o,B)),Uub: $o,Uuc: $o] : aa($o,A,aa($o,fun($o,A),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_be(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),Uu),Uua),(Uub)),(Uuc)) = aa(B,A,Uu,aa($o,B,aa($o,fun($o,B),Uua,(Uub)),(Uuc))) ).

% ATP.lambda_650
tff(fact_8830_ATP_Olambda__651,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_ih(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_651
tff(fact_8831_ATP_Olambda__652,axiom,
    ! [A: $tType,C: $tType,B: $tType,D6: $tType,Uu: fun(B,A),Uua: fun(C,fun(D6,B)),Uub: C,Uuc: D6] : aa(D6,A,aa(C,fun(D6,A),aa(fun(C,fun(D6,B)),fun(C,fun(D6,A)),aTP_Lamp_vk(fun(B,A),fun(fun(C,fun(D6,B)),fun(C,fun(D6,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D6,B,aa(C,fun(D6,B),Uua,Uub),Uuc)) ).

% ATP.lambda_652
tff(fact_8832_ATP_Olambda__653,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_op(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_653
tff(fact_8833_ATP_Olambda__654,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_hp(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_654
tff(fact_8834_ATP_Olambda__655,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_on(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_655
tff(fact_8835_ATP_Olambda__656,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_jj(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_656
tff(fact_8836_ATP_Olambda__657,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: nat,Uub: list(A),Uuc: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_lu(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(A,$o,Uu,aa(nat,A,nth(A,take(A,Uua,Uub)),Uuc)) ) ).

% ATP.lambda_657
tff(fact_8837_ATP_Olambda__658,axiom,
    ! [A: $tType,D6: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D6,B),Uub: D6,Uuc: C] : aa(C,A,aa(D6,fun(C,A),aa(fun(D6,B),fun(D6,fun(C,A)),aTP_Lamp_um(fun(product_prod(B,C),A),fun(fun(D6,B),fun(D6,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D6,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_658
tff(fact_8838_ATP_Olambda__659,axiom,
    ! [A: $tType,B: $tType,C: $tType,D6: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D6,C),Uub: B,Uuc: D6] : aa(D6,A,aa(B,fun(D6,A),aa(fun(D6,C),fun(B,fun(D6,A)),aTP_Lamp_un(fun(product_prod(B,C),A),fun(fun(D6,C),fun(B,fun(D6,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D6,C,Uua,Uuc))) ).

% ATP.lambda_659
tff(fact_8839_ATP_Olambda__660,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_ui(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) = aa(fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),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_uh(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_660
tff(fact_8840_ATP_Olambda__661,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_na(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_mz(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_661
tff(fact_8841_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(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_my(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_mx(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_662
tff(fact_8842_ATP_Olambda__663,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_acc(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = complete_Inf_Inf(set(A),aa(set(C),set(set(A)),image(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_acb(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_663
tff(fact_8843_ATP_Olambda__664,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(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_abw(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = complete_Inf_Inf(A,aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_abv(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_664
tff(fact_8844_ATP_Olambda__665,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yh(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_665
tff(fact_8845_ATP_Olambda__666,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yf(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_666
tff(fact_8846_ATP_Olambda__667,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aah(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,comp($o,$o,A,fNot,Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_667
tff(fact_8847_ATP_Olambda__668,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_ws(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),cons(A,Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_668
tff(fact_8848_ATP_Olambda__669,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yj(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_669
tff(fact_8849_ATP_Olambda__670,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_yl(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_670
tff(fact_8850_ATP_Olambda__671,axiom,
    ! [C: $tType,A: $tType,B: $tType,E4: $tType,D6: $tType,Uu: fun(B,fun(C,fun(D6,fun(E4,set(A))))),Uua: set(product_prod(D6,E4)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(set(product_prod(D6,E4)),fun(B,fun(C,set(A))),aTP_Lamp_uu(fun(B,fun(C,fun(D6,fun(E4,set(A))))),fun(set(product_prod(D6,E4)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D6,E4)),set(set(A)),image(product_prod(D6,E4),set(A),aa(fun(D6,fun(E4,set(A))),fun(product_prod(D6,E4),set(A)),product_case_prod(D6,E4,set(A)),aa(C,fun(D6,fun(E4,set(A))),aa(B,fun(C,fun(D6,fun(E4,set(A)))),Uu,Uub),Uuc))),Uua)) ).

% ATP.lambda_671
tff(fact_8851_ATP_Olambda__672,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_jz(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(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_jy(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_672
tff(fact_8852_ATP_Olambda__673,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu: fun(code_integer,product_prod(A,A)),Uua: A,Uub: code_integer,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_vq(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),Uu),Uua),Uub),Uuc),Uud) = append(A,code_T4080844693773952564amples(A,Uu,Uua,aa(code_integer,code_integer,minus_minus(code_integer,Uub),one_one(code_integer))),aa(list(A),list(A),cons(A,Uuc),aa(list(A),list(A),cons(A,Uud),nil(A)))) ) ).

% ATP.lambda_673
tff(fact_8853_ATP_Olambda__674,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu: fun(code_integer,product_prod(A,A)),Uua: code_integer,Uub: A,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(A,fun(A,fun(A,list(A))),aa(code_integer,fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_vp(fun(code_integer,product_prod(A,A)),fun(code_integer,fun(A,fun(A,fun(A,list(A))))),Uu),Uua),Uub),Uuc),Uud) = append(A,code_T4080844693773952564amples(A,Uu,Uub,aa(code_integer,code_integer,minus_minus(code_integer,Uua),one_one(code_integer))),aa(list(A),list(A),cons(A,Uuc),aa(list(A),list(A),cons(A,Uud),nil(A)))) ) ).

% ATP.lambda_674
tff(fact_8854_ATP_Olambda__675,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_cr(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_675
tff(fact_8855_ATP_Olambda__676,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_jx(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jw(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_676
tff(fact_8856_ATP_Olambda__677,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_mb(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_677
tff(fact_8857_ATP_Olambda__678,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_lz(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_678
tff(fact_8858_ATP_Olambda__679,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_vi(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_trancl(A,Uub))
          | ( Uuc = Uu ) )
        & ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_trancl(A,Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_679
tff(fact_8859_ATP_Olambda__680,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_ma(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,uminus_uminus(A,Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(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_680
tff(fact_8860_ATP_Olambda__681,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_jy(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_681
tff(fact_8861_ATP_Olambda__682,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_cv(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_682
tff(fact_8862_ATP_Olambda__683,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_bw(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_683
tff(fact_8863_ATP_Olambda__684,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_cw(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_684
tff(fact_8864_ATP_Olambda__685,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_uh(fun(a,b),fun(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $let(
        x: b,
        x:= aa(a,b,Uu,Uub),
        $ite(
          aa(b,$o,aa(b,fun(b,$o),ord_less(b),x),Uua),
          aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),aa(list(a),list(a),cons(a,Uub),Uuc)),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),Uue)),
          $ite(aa(b,$o,aa(b,fun(b,$o),ord_less(b),Uua),x),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),aa(list(a),list(a),cons(a,Uub),Uue))),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),aa(list(a),list(a),cons(a,Uub),Uud)),Uue))) ) ) ).

% ATP.lambda_685
tff(fact_8865_ATP_Olambda__686,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,A,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,A),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_bd(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,A,Uu,aa(vEBT_VEBTi,B,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,B),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_686
tff(fact_8866_ATP_Olambda__687,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_de(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_687
tff(fact_8867_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_zg($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_688
tff(fact_8868_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType,Uu: multiset(B),Uua: A] : aa(A,multiset(B),aTP_Lamp_ex(multiset(B),fun(A,multiset(B)),Uu),Uua) = Uu ).

% ATP.lambda_689
tff(fact_8869_ATP_Olambda__690,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_zm(assn,fun(A,assn),Uu),Uua) = Uu ).

% ATP.lambda_690
tff(fact_8870_ATP_Olambda__691,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Uu: word(B),Uua: A] : aa(A,word(B),aTP_Lamp_yx(word(B),fun(A,word(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_691
tff(fact_8871_ATP_Olambda__692,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_zz($o,fun(nat,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_692
tff(fact_8872_ATP_Olambda__693,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_zc($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_693
tff(fact_8873_ATP_Olambda__694,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_ff(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_694
tff(fact_8874_ATP_Olambda__695,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_vj(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).

% ATP.lambda_695
tff(fact_8875_ATP_Olambda__696,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_en(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_696
tff(fact_8876_ATP_Olambda__697,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_fe(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_697
tff(fact_8877_ATP_Olambda__698,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_fd(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_698
tff(fact_8878_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_eo(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_699
tff(fact_8879_ATP_Olambda__700,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_da(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_700
tff(fact_8880_ATP_Olambda__701,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ga(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_701
tff(fact_8881_ATP_Olambda__702,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ro(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_702
tff(fact_8882_ATP_Olambda__703,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_fx(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_703
tff(fact_8883_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aar(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_704
tff(fact_8884_ATP_Olambda__705,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aaj(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_705
tff(fact_8885_ATP_Olambda__706,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_yn(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_706
tff(fact_8886_ATP_Olambda__707,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ie(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_707
tff(fact_8887_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_mc(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_708
tff(fact_8888_ATP_Olambda__709,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lg(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_709
tff(fact_8889_ATP_Olambda__710,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_bx(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_710
tff(fact_8890_ATP_Olambda__711,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ls(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_711
tff(fact_8891_ATP_Olambda__712,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_jb(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_712
tff(fact_8892_ATP_Olambda__713,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_ja(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_713
tff(fact_8893_ATP_Olambda__714,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_vs(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_714
tff(fact_8894_ATP_Olambda__715,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_br(A,fun(A,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_715
tff(fact_8895_ATP_Olambda__716,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_hw(complex,complex),Uu) = Uu ).

% ATP.lambda_716
tff(fact_8896_ATP_Olambda__717,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A)] : aa(word(A),word(A),aTP_Lamp_aad(word(A),word(A)),Uu) = Uu ) ).

% ATP.lambda_717
tff(fact_8897_ATP_Olambda__718,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_kx(nat,nat),Uu) = Uu ).

% ATP.lambda_718
tff(fact_8898_ATP_Olambda__719,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_ph(int,int),Uu) = Uu ).

% ATP.lambda_719
tff(fact_8899_ATP_Olambda__720,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_act(B,B),Uu) = Uu ) ).

% ATP.lambda_720
tff(fact_8900_ATP_Olambda__721,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_acs(B,B),Uu) = Uu ) ).

% ATP.lambda_721
tff(fact_8901_ATP_Olambda__722,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ib(A,A),Uu) = Uu ) ).

% ATP.lambda_722
tff(fact_8902_ATP_Olambda__723,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ez(A,A),Uu) = Uu ) ).

% ATP.lambda_723
tff(fact_8903_ATP_Olambda__724,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_abg(A,A),Uu) = Uu ) ).

% ATP.lambda_724
tff(fact_8904_ATP_Olambda__725,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_dw(A,A),Uu) = Uu ) ).

% ATP.lambda_725
tff(fact_8905_ATP_Olambda__726,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_az(A,A),Uu) = Uu ) ).

% ATP.lambda_726
tff(fact_8906_ATP_Olambda__727,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_bp(A,A),Uu) = Uu ).

% ATP.lambda_727
tff(fact_8907_ATP_Olambda__728,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_abj(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_728
tff(fact_8908_ATP_Olambda__729,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_abi(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_729
tff(fact_8909_ATP_Olambda__730,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_aae(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_730
tff(fact_8910_ATP_Olambda__731,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_fs(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_731
tff(fact_8911_ATP_Olambda__732,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_fc(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_732
tff(fact_8912_ATP_Olambda__733,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_fb(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_733
tff(fact_8913_ATP_Olambda__734,axiom,
    ! [A: $tType,D6: $tType,Uu: A] : aa(A,set(D6),aTP_Lamp_aaf(A,set(D6)),Uu) = bot_bot(set(D6)) ).

% ATP.lambda_734
tff(fact_8914_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_qx(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_735
tff(fact_8915_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_qr(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_736
tff(fact_8916_ATP_Olambda__737,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_gj(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_737
tff(fact_8917_ATP_Olambda__738,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_hz(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_738
tff(fact_8918_ATP_Olambda__739,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ac(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_739
tff(fact_8919_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_yr(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_740
tff(fact_8920_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_nn(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_741
tff(fact_8921_ATP_Olambda__742,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_aas(A,real),Uu) = one_one(real) ).

% ATP.lambda_742
tff(fact_8922_ATP_Olambda__743,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_aam(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_743
tff(fact_8923_ATP_Olambda__744,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_bu(B,option(A)),Uu) = none(A) ).

% ATP.lambda_744
tff(fact_8924_ATP_Olambda__745,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_am(A,option(B)),Uu) = none(B) ).

% ATP.lambda_745
tff(fact_8925_ATP_Olambda__746,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_wk(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_746
tff(fact_8926_ATP_Olambda__747,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_xs(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_747
tff(fact_8927_ATP_Olambda__748,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_aj(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_748
tff(fact_8928_ATP_Olambda__749,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_xr(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_749
tff(fact_8929_ATP_Olambda__750,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_wz(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_750
tff(fact_8930_ATP_Olambda__751,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_bl(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_751

% Type constructors (1191)
tff(tcon_VEBT__BuildupMemImp_OVEBTi___Typerep_Otyperep,axiom,
    typerep(vEBT_VEBTi) ).

tff(tcon_VEBT__Definitions_OVEBT___Typerep_Otyperep_1,axiom,
    typerep(vEBT_VEBT) ).

tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_2,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(heap_Time_Heap(A14)) ) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_3,axiom,
    typerep(code_integer) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_4,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(heap_ext(A14)) ) ).

tff(tcon_Product__Type_Ounit___Typerep_Otyperep_5,axiom,
    typerep(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_6,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => typerep(product_prod(A14,A15)) ) ).

tff(tcon_Numeral__Type_Onum0___Typerep_Otyperep_7,axiom,
    typerep(numeral_num0) ).

tff(tcon_Numeral__Type_Obit1___Typerep_Otyperep_8,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Typerep_Otyperep_9,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(numeral_bit0(A14)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_10,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(multiset(A14)) ) ).

tff(tcon_Extended__Nat_Oenat___Typerep_Otyperep_11,axiom,
    typerep(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_12,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Complex_Ocomplex___Typerep_Otyperep_13,axiom,
    typerep(complex) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_14,axiom,
    typerep(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_15,axiom,
    bounded_lattice_top(assn) ).

tff(tcon_String_Oliteral___Typerep_Otyperep_16,axiom,
    typerep(literal) ).

tff(tcon_Uint32_Ouint32___Typerep_Otyperep_17,axiom,
    typerep(uint32) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_18,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_19,axiom,
    ! [A14: $tType] :
      ( bounded_lattice_top(A14)
     => bounded_lattice_top(option(A14)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_20,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(filter(A14)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_21,axiom,
    ! [A14: $tType] : bounded_lattice_top(filter(A14)) ).

tff(tcon_Sum__Type_Osum___Typerep_Otyperep_22,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => typerep(sum_sum(A14,A15)) ) ).

tff(tcon_Heap_Oarray___Typerep_Otyperep_23,axiom,
    ! [A14: $tType] : typerep(array(A14)) ).

tff(tcon_Word_Oword___Typerep_Otyperep_24,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(word(A14)) ) ).

tff(tcon_Real_Oreal___Typerep_Otyperep_25,axiom,
    typerep(real) ).

tff(tcon_List_Olist___Typerep_Otyperep_26,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(list(A14)) ) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_27,axiom,
    typerep($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_28,axiom,
    bounded_lattice_top($o) ).

tff(tcon_Set_Oset___Typerep_Otyperep_29,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => typerep(set(A14)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_30,axiom,
    ! [A14: $tType] : bounded_lattice_top(set(A14)) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_31,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Typerep_Otyperep_32,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_33,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Typerep_Otyperep_34,axiom,
    typerep(int) ).

tff(tcon_fun___Typerep_Otyperep_35,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => typerep(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top_36,axiom,
    ! [A14: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounded_lattice_top(fun(A14,A15)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A14: $tType,A15: $tType] :
      ( comple6319245703460814977attice(A15)
     => condit1219197933456340205attice(fun(A14,A15)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A14: $tType,A15: $tType] :
      ( counta3822494911875563373attice(A15)
     => counta3822494911875563373attice(fun(A14,A15)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A14: $tType,A15: $tType] :
      ( comple592849572758109894attice(A15)
     => comple592849572758109894attice(fun(A14,A15)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A14: $tType,A15: $tType] :
      ( comple489889107523837845lgebra(A15)
     => comple489889107523837845lgebra(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A14: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounde4967611905675639751up_bot(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A14: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounde4346867609351753570nf_top(fun(A14,A15)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A14: $tType,A15: $tType] :
      ( comple6319245703460814977attice(A15)
     => comple6319245703460814977attice(fun(A14,A15)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A14: $tType,A15: $tType] :
      ( boolea8198339166811842893lgebra(A15)
     => boolea8198339166811842893lgebra(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A14: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounded_lattice_bot(fun(A14,A15)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A14: $tType,A15: $tType] :
      ( comple6319245703460814977attice(A15)
     => comple9053668089753744459l_ccpo(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A14: $tType,A15: $tType] :
      ( semilattice_sup(A15)
     => semilattice_sup(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A14: $tType,A15: $tType] :
      ( semilattice_inf(A15)
     => semilattice_inf(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A14: $tType,A15: $tType] :
      ( bounded_lattice(A15)
     => bounded_lattice(fun(A14,A15)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => code_term_of(fun(A14,A15)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A14: $tType,A15: $tType] :
      ( complete_Sup(A15)
     => complete_Sup(fun(A14,A15)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A14: $tType,A15: $tType] :
      ( complete_Inf(A15)
     => complete_Inf(fun(A14,A15)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A14: $tType,A15: $tType] :
      ( order_top(A15)
     => order_top(fun(A14,A15)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A14: $tType,A15: $tType] :
      ( order_bot(A15)
     => order_bot(fun(A14,A15)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A14: $tType,A15: $tType] :
      ( preorder(A15)
     => preorder(fun(A14,A15)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( finite_finite(A14)
        & finite_finite(A15) )
     => finite_finite(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A14: $tType,A15: $tType] :
      ( lattice(A15)
     => lattice(fun(A14,A15)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A14: $tType,A15: $tType] :
      ( order(A15)
     => order(fun(A14,A15)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ord(A15)
     => ord(fun(A14,A15)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A14: $tType,A15: $tType] :
      ( bot(A15)
     => bot(fun(A14,A15)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A14: $tType,A15: $tType] :
      ( uminus(A15)
     => uminus(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Osup,axiom,
    ! [A14: $tType,A15: $tType] :
      ( semilattice_sup(A15)
     => sup(fun(A14,A15)) ) ).

tff(tcon_fun___Lattices_Oinf,axiom,
    ! [A14: $tType,A15: $tType] :
      ( semilattice_inf(A15)
     => inf(fun(A14,A15)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A14: $tType,A15: $tType] :
      ( minus(A15)
     => minus(fun(A14,A15)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_37,axiom,
    condit1219197933456340205attice(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___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
    quickc6926020345158392990erm_of(int) ).

tff(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
    bit_bi6583157726757044596ension(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(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___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_38,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_39,axiom,
    semilattice_inf(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___Code__Evaluation_Oterm__of_40,axiom,
    code_term_of(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OSup_41,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_42,axiom,
    complete_Inf(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___Orderings_Opreorder_43,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___Lattices_Olattice_44,axiom,
    lattice(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_45,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_46,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_47,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___Lattices_Osup_48,axiom,
    sup(int) ).

tff(tcon_Int_Oint___Lattices_Oinf_49,axiom,
    inf(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___Groups_Ominus_50,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_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_51,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_52,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_53,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_54,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_55,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_56,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_57,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_58,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_59,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_60,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_61,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_62,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_63,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_64,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_65,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_66,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_67,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_68,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_69,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_70,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_71,axiom,
    quickc6926020345158392990erm_of(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_72,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_73,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_74,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_75,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_76,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_77,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_78,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_79,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_80,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_81,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_82,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_83,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_84,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_85,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_86,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_87,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_88,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_89,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_90,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_91,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_92,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_93,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_94,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_95,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_96,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_97,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_98,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_99,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_100,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_101,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_102,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_103,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_104,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_105,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_106,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_107,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_108,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_109,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_110,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_111,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_112,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_113,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_114,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_115,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_116,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_117,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_118,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_119,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_120,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_121,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_122,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_123,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_124,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_125,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_126,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_127,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_128,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_129,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_130,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Lattices_Osup_131,axiom,
    sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Oinf_132,axiom,
    inf(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_133,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_134,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_135,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_136,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_137,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_138,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_139,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_140,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_141,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_142,axiom,
    quickc6926020345158392990erm_of(num) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_143,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_144,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_145,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_146,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_147,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_148,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_149,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_150,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_151,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_152,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_153,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_154,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_155,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_156,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_157,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_158,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_159,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_160,axiom,
    quickc6926020345158392990erm_of(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_161,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_162,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_163,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_164,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_165,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_166,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_167,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_168,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_169,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_170,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_171,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_172,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_173,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_174,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_175,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_176,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_177,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_178,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_179,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_180,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_181,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_182,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_183,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_184,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_185,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_186,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_187,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_188,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_189,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_190,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_191,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_192,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_193,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_194,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_195,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_196,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_197,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_198,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_199,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_200,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_201,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_202,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_203,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_204,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_205,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_206,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_207,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_208,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_209,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_210,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_211,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_212,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_213,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_214,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_215,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_216,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_217,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_218,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_219,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_220,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_221,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_222,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_223,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_224,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_225,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_226,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_227,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_228,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_229,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_230,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Lattices_Osup_231,axiom,
    sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Oinf_232,axiom,
    inf(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_233,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_234,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_235,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_236,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_237,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_238,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_239,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_240,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_241,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_242,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_243,axiom,
    ! [A14: $tType] : condit1219197933456340205attice(set(A14)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_244,axiom,
    ! [A14: $tType] : counta3822494911875563373attice(set(A14)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_245,axiom,
    ! [A14: $tType] : comple592849572758109894attice(set(A14)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_246,axiom,
    ! [A14: $tType] : comple489889107523837845lgebra(set(A14)) ).

tff(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_247,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(set(A14)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_248,axiom,
    ! [A14: $tType] : bounde4967611905675639751up_bot(set(A14)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_249,axiom,
    ! [A14: $tType] : bounde4346867609351753570nf_top(set(A14)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_250,axiom,
    ! [A14: $tType] : comple6319245703460814977attice(set(A14)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_251,axiom,
    ! [A14: $tType] : boolea8198339166811842893lgebra(set(A14)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_252,axiom,
    ! [A14: $tType] : bounded_lattice_bot(set(A14)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_253,axiom,
    ! [A14: $tType] : comple9053668089753744459l_ccpo(set(A14)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_254,axiom,
    ! [A14: $tType] : semilattice_sup(set(A14)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_255,axiom,
    ! [A14: $tType] : semilattice_inf(set(A14)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_256,axiom,
    ! [A14: $tType] : bounded_lattice(set(A14)) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_257,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(set(A14)) ) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_258,axiom,
    ! [A14: $tType] : complete_Sup(set(A14)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_259,axiom,
    ! [A14: $tType] : complete_Inf(set(A14)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_260,axiom,
    ! [A14: $tType] : order_top(set(A14)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_261,axiom,
    ! [A14: $tType] : order_bot(set(A14)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_262,axiom,
    ! [A14: $tType] : preorder(set(A14)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_263,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => finite_finite(set(A14)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_264,axiom,
    ! [A14: $tType] : lattice(set(A14)) ).

tff(tcon_Set_Oset___Orderings_Oorder_265,axiom,
    ! [A14: $tType] : order(set(A14)) ).

tff(tcon_Set_Oset___Orderings_Oord_266,axiom,
    ! [A14: $tType] : ord(set(A14)) ).

tff(tcon_Set_Oset___Orderings_Obot_267,axiom,
    ! [A14: $tType] : bot(set(A14)) ).

tff(tcon_Set_Oset___Groups_Ouminus_268,axiom,
    ! [A14: $tType] : uminus(set(A14)) ).

tff(tcon_Set_Oset___Lattices_Osup_269,axiom,
    ! [A14: $tType] : sup(set(A14)) ).

tff(tcon_Set_Oset___Lattices_Oinf_270,axiom,
    ! [A14: $tType] : inf(set(A14)) ).

tff(tcon_Set_Oset___Groups_Ominus_271,axiom,
    ! [A14: $tType] : minus(set(A14)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_272,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_273,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_274,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_275,axiom,
    comple489889107523837845lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_276,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_277,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_278,axiom,
    quickc6926020345158392990erm_of($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_279,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_280,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_281,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_282,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_283,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_284,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_285,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_286,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_287,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_288,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_289,axiom,
    code_term_of($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_290,axiom,
    complete_Sup($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_291,axiom,
    complete_Inf($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_292,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_293,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_294,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_295,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_296,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_297,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_298,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_299,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_300,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_301,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Lattices_Osup_302,axiom,
    sup($o) ).

tff(tcon_HOL_Obool___Lattices_Oinf_303,axiom,
    inf($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_304,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_305,axiom,
    heap($o) ).

tff(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_306,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(list(A14)) ) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_307,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(list(A14)) ) ).

tff(tcon_List_Olist___Heap_Oheap_308,axiom,
    ! [A14: $tType] :
      ( heap(A14)
     => heap(list(A14)) ) ).

tff(tcon_List_Olist___Nat_Osize_309,axiom,
    ! [A14: $tType] : size(list(A14)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_310,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_311,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_312,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
    condit5016429287641298734tinuum(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_313,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_314,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_315,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_316,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_317,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_318,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_319,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_320,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_321,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_322,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_323,axiom,
    quickc6926020345158392990erm_of(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_324,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_325,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_326,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_327,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_328,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_329,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_330,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_331,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_332,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_333,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_334,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_335,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_336,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_337,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_338,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_339,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_340,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_341,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_342,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_343,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_344,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_345,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_346,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_347,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_348,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_349,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_350,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_351,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_352,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_353,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_354,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_355,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_356,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_357,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Code__Evaluation_Oterm__of_358,axiom,
    code_term_of(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_359,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_360,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_361,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_362,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_363,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_364,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_365,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_366,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_367,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Complete__Lattices_OSup_368,axiom,
    complete_Sup(real) ).

tff(tcon_Real_Oreal___Complete__Lattices_OInf_369,axiom,
    complete_Inf(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_370,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_371,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_372,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_373,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_374,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_375,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_376,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_377,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_378,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_379,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_380,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_381,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_382,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_383,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_384,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_385,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_386,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_387,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_388,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_389,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_390,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_391,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_392,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_393,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_394,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_395,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_396,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_397,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_398,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_399,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_400,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_401,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_402,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_403,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_404,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Lattices_Osup_405,axiom,
    sup(real) ).

tff(tcon_Real_Oreal___Lattices_Oinf_406,axiom,
    inf(real) ).

tff(tcon_Real_Oreal___Groups_Otimes_407,axiom,
    times(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_408,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_409,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_410,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_411,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_412,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_413,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_414,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_415,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_416,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_417,axiom,
    dvd(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_418,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => bit_se359711467146920520ations(word(A14)) ) ).

tff(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_419,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(word(A14)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_420,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => bit_bi6583157726757044596ension(word(A14)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_421,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => bit_ri3973907225187159222ations(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_422,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => cancel2418104881723323429up_add(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_423,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => cancel1802427076303600483id_add(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_424,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_s4317794764714335236cancel(word(A14)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_425,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => bit_semiring_bits(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_426,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => cancel_semigroup_add(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_427,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => ab_semigroup_mult(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_428,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semiring_1_cancel(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_429,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_monoid_mult(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_430,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => ab_semigroup_add(word(A14)) ) ).

tff(tcon_Word_Oword___Code__Evaluation_Oterm__of_431,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(word(A14)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_432,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semiring_parity(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_433,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_monoid_add(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_434,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semiring_modulo(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_435,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_semiring_1(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_436,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_semiring_0(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_437,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semigroup_mult(word(A14)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_438,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semiring_numeral(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_439,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semigroup_add(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_440,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_semiring(word(A14)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_441,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => wellorder(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_442,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => ab_group_add(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_443,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => zero_neq_one(word(A14)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_444,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => preorder(word(A14)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_445,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => linorder(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_446,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => monoid_mult(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_447,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_ring_1(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_448,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => monoid_add(word(A14)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_449,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => finite_finite(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_450,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semiring_1(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_451,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semiring_0(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_452,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => group_add(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_453,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => mult_zero(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_454,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => comm_ring(word(A14)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_455,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => order(word(A14)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_456,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => neg_numeral(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_457,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => semiring(word(A14)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_458,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => ord(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_459,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => uminus(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_460,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => ring_1(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Otimes_461,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => times(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ominus_462,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => minus(word(A14)) ) ).

tff(tcon_Word_Oword___Power_Opower_463,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => power(word(A14)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_464,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => numeral(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_465,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => zero(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_466,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => plus(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Oring_467,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => ring(word(A14)) ) ).

tff(tcon_Word_Oword___Groups_Oone_468,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => one(word(A14)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_469,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => dvd(word(A14)) ) ).

tff(tcon_Word_Oword___Nat_Osize_470,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => size(word(A14)) ) ).

tff(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_471,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(array(A14)) ) ).

tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_472,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(array(A14)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_473,axiom,
    ! [A14: $tType] : heap(array(A14)) ).

tff(tcon_Heap_Oarray___Nat_Osize_474,axiom,
    ! [A14: $tType] : size(array(A14)) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Narrowing_Opartial__term__of_475,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => quickc6926020345158392990erm_of(sum_sum(A14,A15)) ) ).

tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_476,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => code_term_of(sum_sum(A14,A15)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_477,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( finite_finite(A14)
        & finite_finite(A15) )
     => finite_finite(sum_sum(A14,A15)) ) ).

tff(tcon_Sum__Type_Osum___Heap_Oheap_478,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( heap(A14)
        & heap(A15) )
     => heap(sum_sum(A14,A15)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_479,axiom,
    ! [A14: $tType,A15: $tType] : size(sum_sum(A14,A15)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_480,axiom,
    ! [A14: $tType] : condit1219197933456340205attice(filter(A14)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_481,axiom,
    ! [A14: $tType] : counta3822494911875563373attice(filter(A14)) ).

tff(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_482,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(filter(A14)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_483,axiom,
    ! [A14: $tType] : bounde4967611905675639751up_bot(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_484,axiom,
    ! [A14: $tType] : bounde4346867609351753570nf_top(filter(A14)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_485,axiom,
    ! [A14: $tType] : comple6319245703460814977attice(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_486,axiom,
    ! [A14: $tType] : bounded_lattice_bot(filter(A14)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_487,axiom,
    ! [A14: $tType] : comple9053668089753744459l_ccpo(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_488,axiom,
    ! [A14: $tType] : semilattice_sup(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_489,axiom,
    ! [A14: $tType] : semilattice_inf(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_490,axiom,
    ! [A14: $tType] : bounded_lattice(filter(A14)) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_491,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(filter(A14)) ) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_492,axiom,
    ! [A14: $tType] : complete_Sup(filter(A14)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_493,axiom,
    ! [A14: $tType] : complete_Inf(filter(A14)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_494,axiom,
    ! [A14: $tType] : order_top(filter(A14)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_495,axiom,
    ! [A14: $tType] : order_bot(filter(A14)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_496,axiom,
    ! [A14: $tType] : preorder(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_497,axiom,
    ! [A14: $tType] : lattice(filter(A14)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_498,axiom,
    ! [A14: $tType] : order(filter(A14)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_499,axiom,
    ! [A14: $tType] : ord(filter(A14)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_500,axiom,
    ! [A14: $tType] : bot(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Osup_501,axiom,
    ! [A14: $tType] : sup(filter(A14)) ).

tff(tcon_Filter_Ofilter___Lattices_Oinf_502,axiom,
    ! [A14: $tType] : inf(filter(A14)) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_503,axiom,
    ! [A14: $tType] :
      ( comple5582772986160207858norder(A14)
     => condit6923001295902523014norder(option(A14)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_504,axiom,
    ! [A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => condit1219197933456340205attice(option(A14)) ) ).

tff(tcon_Option_Ooption___Countable__Complete__Lattices_Ocountable__complete__lattice_505,axiom,
    ! [A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => counta3822494911875563373attice(option(A14)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_506,axiom,
    ! [A14: $tType] :
      ( comple592849572758109894attice(A14)
     => comple592849572758109894attice(option(A14)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_507,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_508,axiom,
    ! [A14: $tType] :
      ( lattice(A14)
     => bounde4967611905675639751up_bot(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_509,axiom,
    ! [A14: $tType] :
      ( bounded_lattice_top(A14)
     => bounde4346867609351753570nf_top(option(A14)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
    ! [A14: $tType] :
      ( comple5582772986160207858norder(A14)
     => comple5582772986160207858norder(option(A14)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_510,axiom,
    ! [A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => comple6319245703460814977attice(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_511,axiom,
    ! [A14: $tType] :
      ( lattice(A14)
     => bounded_lattice_bot(option(A14)) ) ).

tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_512,axiom,
    ! [A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => comple9053668089753744459l_ccpo(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_513,axiom,
    ! [A14: $tType] :
      ( semilattice_sup(A14)
     => semilattice_sup(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_514,axiom,
    ! [A14: $tType] :
      ( semilattice_inf(A14)
     => semilattice_inf(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_515,axiom,
    ! [A14: $tType] :
      ( bounded_lattice_top(A14)
     => bounded_lattice(option(A14)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_516,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(option(A14)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OSup_517,axiom,
    ! [A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => complete_Sup(option(A14)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OInf_518,axiom,
    ! [A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => complete_Inf(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_519,axiom,
    ! [A14: $tType] :
      ( wellorder(A14)
     => wellorder(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_520,axiom,
    ! [A14: $tType] :
      ( order_top(A14)
     => order_top(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_521,axiom,
    ! [A14: $tType] :
      ( order(A14)
     => order_bot(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_522,axiom,
    ! [A14: $tType] :
      ( preorder(A14)
     => preorder(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_523,axiom,
    ! [A14: $tType] :
      ( linorder(A14)
     => linorder(option(A14)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_524,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => finite_finite(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_525,axiom,
    ! [A14: $tType] :
      ( lattice(A14)
     => lattice(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_526,axiom,
    ! [A14: $tType] :
      ( order(A14)
     => order(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_527,axiom,
    ! [A14: $tType] :
      ( preorder(A14)
     => ord(option(A14)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_528,axiom,
    ! [A14: $tType] :
      ( order(A14)
     => bot(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Osup_529,axiom,
    ! [A14: $tType] :
      ( sup(A14)
     => sup(option(A14)) ) ).

tff(tcon_Option_Ooption___Lattices_Oinf_530,axiom,
    ! [A14: $tType] :
      ( inf(A14)
     => inf(option(A14)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_531,axiom,
    ! [A14: $tType] :
      ( heap(A14)
     => heap(option(A14)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_532,axiom,
    ! [A14: $tType] : size(option(A14)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_533,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_534,axiom,
    quickc6926020345158392990erm_of(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_535,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_536,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_537,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_538,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_539,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_540,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_541,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_542,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_543,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_544,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_545,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_546,axiom,
    code_term_of(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_547,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_548,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_549,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_550,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_551,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_552,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_553,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_554,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_555,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_556,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_557,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_558,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_559,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_560,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_561,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_562,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_563,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_564,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_565,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_566,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_567,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_568,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_569,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_570,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_571,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_572,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_573,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Otimes_574,axiom,
    times(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ominus_575,axiom,
    minus(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_576,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_577,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_578,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_579,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_580,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_581,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_582,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_583,axiom,
    size(uint32) ).

tff(tcon_String_Oliteral___Quickcheck__Narrowing_Opartial__term__of_584,axiom,
    quickc6926020345158392990erm_of(literal) ).

tff(tcon_String_Oliteral___Code__Evaluation_Oterm__of_585,axiom,
    code_term_of(literal) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_586,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_587,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_588,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_589,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_590,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_591,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_592,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_593,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Heap_Oheap_594,axiom,
    heap(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_595,axiom,
    size(literal) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_596,axiom,
    bounde4967611905675639751up_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_597,axiom,
    bounde4346867609351753570nf_top(assn) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_598,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_599,axiom,
    bounded_lattice_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_600,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_601,axiom,
    semilattice_inf(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_602,axiom,
    bounded_lattice(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_603,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_604,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_605,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_606,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_607,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_608,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_609,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_610,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_611,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_612,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_613,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_614,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osup_615,axiom,
    sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Oinf_616,axiom,
    inf(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_617,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_618,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_619,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_620,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_621,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_622,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_623,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_624,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_625,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_626,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_627,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_628,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_629,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_630,axiom,
    quickc6926020345158392990erm_of(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_631,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_632,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_633,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_634,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_635,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_636,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_637,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_638,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_639,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_640,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_641,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_642,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_643,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_644,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_645,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_646,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_647,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_648,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_649,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_650,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_651,axiom,
    code_term_of(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_652,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_653,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_654,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_655,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_656,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_657,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_658,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_659,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_660,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_661,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_662,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_663,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_664,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_665,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_666,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_667,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_668,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_669,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_670,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_671,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_672,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_673,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_674,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_675,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_676,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_677,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_678,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_679,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_680,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Otimes_681,axiom,
    times(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_682,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_683,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_684,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_685,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_686,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_687,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_688,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_689,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_690,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_691,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_692,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_693,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_694,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_695,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_696,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_697,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_698,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_699,axiom,
    quickc6926020345158392990erm_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_700,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_701,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_702,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_703,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_704,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_705,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_706,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_707,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_708,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_709,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_710,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_711,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_712,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_713,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_714,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_715,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_716,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_717,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_718,axiom,
    code_term_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_719,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_720,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_721,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_722,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_723,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_OSup_724,axiom,
    complete_Sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_OInf_725,axiom,
    complete_Inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_726,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_727,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_728,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_729,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_730,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_731,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_732,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_733,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_734,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_735,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_736,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_737,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_738,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_739,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_740,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_741,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_742,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_743,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_744,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_745,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_746,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osup_747,axiom,
    sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Oinf_748,axiom,
    inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Otimes_749,axiom,
    times(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_750,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_751,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_752,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_753,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_754,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_755,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_756,axiom,
    dvd(extended_enat) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_757,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(multiset(A14)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_758,axiom,
    ! [A14: $tType] :
      ( preorder(A14)
     => ordere6658533253407199908up_add(multiset(A14)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_759,axiom,
    ! [A14: $tType] : cancel2418104881723323429up_add(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_760,axiom,
    ! [A14: $tType] : cancel1802427076303600483id_add(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_761,axiom,
    ! [A14: $tType] : cancel_semigroup_add(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_762,axiom,
    ! [A14: $tType] : comm_monoid_diff(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_763,axiom,
    ! [A14: $tType] : ab_semigroup_add(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_764,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(multiset(A14)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_765,axiom,
    ! [A14: $tType] : comm_monoid_add(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OSup_766,axiom,
    ! [A14: $tType] : complete_Sup(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OInf_767,axiom,
    ! [A14: $tType] : complete_Inf(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_768,axiom,
    ! [A14: $tType] : semigroup_add(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_769,axiom,
    ! [A14: $tType] :
      ( preorder(A14)
     => preorder(multiset(A14)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_770,axiom,
    ! [A14: $tType] : monoid_add(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_771,axiom,
    ! [A14: $tType] :
      ( preorder(A14)
     => order(multiset(A14)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_772,axiom,
    ! [A14: $tType] :
      ( preorder(A14)
     => ord(multiset(A14)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_773,axiom,
    ! [A14: $tType] : minus(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_774,axiom,
    ! [A14: $tType] : zero(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_775,axiom,
    ! [A14: $tType] : plus(multiset(A14)) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_776,axiom,
    ! [A14: $tType] : size(multiset(A14)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_777,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => cancel2418104881723323429up_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_778,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => cancel1802427076303600483id_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_779,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_s4317794764714335236cancel(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_780,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => cancel_semigroup_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_781,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ab_semigroup_mult(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_782,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_1_cancel(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_783,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_monoid_mult(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_784,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ab_semigroup_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_785,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_786,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_monoid_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_787,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_semiring_1(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_788,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_semiring_0(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_789,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semigroup_mult(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_790,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_numeral(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_791,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semigroup_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_792,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_semiring(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_793,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => wellorder(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_794,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ab_group_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_795,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => zero_neq_one(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_796,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => preorder(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_797,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => linorder(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_798,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => monoid_mult(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_799,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_ring_1(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_800,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => monoid_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_801,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => finite_finite(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => card2(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0,axiom,
    ! [A14: $tType] :
      ( type_len0(A14)
     => type_len0(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_802,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_1(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_803,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_0(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_804,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => group_add(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen,axiom,
    ! [A14: $tType] :
      ( type_len(A14)
     => type_len(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_805,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => mult_zero(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_806,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_ring(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_807,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => order(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_808,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => neg_numeral(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_809,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_810,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ord(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_811,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => uminus(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_812,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ring_1(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Otimes_813,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => times(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ominus_814,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => minus(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_815,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => power(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_816,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => numeral(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_817,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => zero(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_818,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => plus(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_819,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ring(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_820,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => one(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_821,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => dvd(numeral_bit0(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_822,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => cancel2418104881723323429up_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_823,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => cancel1802427076303600483id_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_824,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_s4317794764714335236cancel(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_825,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => cancel_semigroup_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_826,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ab_semigroup_mult(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_827,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_1_cancel(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_828,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_monoid_mult(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_829,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ab_semigroup_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_830,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_831,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_monoid_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_832,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_semiring_1(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_833,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_semiring_0(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_834,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semigroup_mult(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_835,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_numeral(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_836,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semigroup_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_837,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_semiring(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_838,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => wellorder(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_839,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ab_group_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_840,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => zero_neq_one(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_841,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => preorder(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_842,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => linorder(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_843,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => monoid_mult(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_844,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_ring_1(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_845,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => monoid_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_846,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => finite_finite(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_847,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => card2(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_848,axiom,
    ! [A14: $tType] :
      ( type_len0(A14)
     => type_len0(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_849,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_1(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_850,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring_0(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_851,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => group_add(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_852,axiom,
    ! [A14: $tType] :
      ( type_len0(A14)
     => type_len(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_853,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => mult_zero(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_854,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => comm_ring(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_855,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => order(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_856,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => neg_numeral(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_857,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => semiring(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_858,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ord(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_859,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => uminus(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_860,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ring_1(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Otimes_861,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => times(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ominus_862,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => minus(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_863,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => power(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_864,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => numeral(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_865,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => zero(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_866,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => plus(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_867,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => ring(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_868,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => one(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_869,axiom,
    ! [A14: $tType] :
      ( finite_finite(A14)
     => dvd(numeral_bit1(A14)) ) ).

tff(tcon_Numeral__Type_Onum0___Quickcheck__Narrowing_Opartial__term__of_870,axiom,
    quickc6926020345158392990erm_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Code__Evaluation_Oterm__of_871,axiom,
    code_term_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Type__Length_Olen0_872,axiom,
    type_len0(numeral_num0) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_873,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( topolo4958980785337419405_space(A14)
        & topolo4958980785337419405_space(A15) )
     => topolo4958980785337419405_space(product_prod(A14,A15)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_874,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => quickc6926020345158392990erm_of(product_prod(A14,A15)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_875,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( topological_t2_space(A14)
        & topological_t2_space(A15) )
     => topological_t2_space(product_prod(A14,A15)) ) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_876,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( typerep(A14)
        & typerep(A15) )
     => code_term_of(product_prod(A14,A15)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_877,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( finite_finite(A14)
        & finite_finite(A15) )
     => finite_finite(product_prod(A14,A15)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_878,axiom,
    ! [A14: $tType,A15: $tType] :
      ( ( heap(A14)
        & heap(A15) )
     => heap(product_prod(A14,A15)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_879,axiom,
    ! [A14: $tType,A15: $tType] : size(product_prod(A14,A15)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_880,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_881,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_882,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_883,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_884,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_885,axiom,
    quickc6926020345158392990erm_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_886,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_887,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_888,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_889,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_890,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_891,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_892,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_893,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_894,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_895,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_896,axiom,
    code_term_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_897,axiom,
    complete_Sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_898,axiom,
    complete_Inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_899,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_900,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_901,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_902,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_903,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_904,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_905,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_906,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_907,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_908,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_909,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osup_910,axiom,
    sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Oinf_911,axiom,
    inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_912,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_913,axiom,
    heap(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_914,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(heap_ext(A14)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_915,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(heap_ext(A14)) ) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_916,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_917,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_918,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_919,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_920,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_921,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_922,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_923,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_924,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_925,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_926,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_927,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_928,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_929,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_930,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_931,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_932,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_933,axiom,
    quickc6926020345158392990erm_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_934,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_935,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_936,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_937,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_938,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_939,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_940,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_941,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_942,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_943,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_944,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_945,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_946,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_947,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_948,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_949,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_950,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_951,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_952,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_953,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_954,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_955,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_956,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_957,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_958,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_959,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_960,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_961,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_962,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_963,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_964,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_965,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_966,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_967,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_968,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_969,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_970,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_971,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_972,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_973,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_974,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_975,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_976,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_977,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_978,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_979,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_980,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_981,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_982,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_983,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_984,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_985,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_986,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_987,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_988,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_989,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_990,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_991,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_992,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_993,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_994,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_995,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_996,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_997,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_998,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_999,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1000,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1001,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1002,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1003,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1004,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1005,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_1006,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_1007,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1008,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1009,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1010,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1011,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1012,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1013,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1014,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1015,axiom,
    dvd(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1016,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => quickc6926020345158392990erm_of(heap_Time_Heap(A14)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1017,axiom,
    ! [A14: $tType] :
      ( typerep(A14)
     => code_term_of(heap_Time_Heap(A14)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1018,axiom,
    ! [A14: $tType] : size(heap_Time_Heap(A14)) ).

tff(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1019,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1020,axiom,
    code_term_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1021,axiom,
    size(vEBT_VEBT) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1022,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1023,axiom,
    code_term_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1024,axiom,
    heap(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1025,axiom,
    size(vEBT_VEBTi) ).

% Helper facts (10)
tff(help_fNot_2_1_U,axiom,
    ! [P: $o] :
      ( (P)
      | aa($o,$o,fNot,(P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: $o] :
      ( ~ aa($o,$o,fNot,(P))
      | ~ (P) ) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: $o,Q: $o] :
      ( ~ aa($o,$o,aa($o,fun($o,$o),fconj,(P)),(Q))
      | (Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: $o,Q: $o] :
      ( ~ aa($o,$o,aa($o,fun($o,$o),fconj,(P)),(Q))
      | (P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: $o,Q: $o] :
      ( ~ (P)
      | ~ (Q)
      | aa($o,$o,aa($o,fun($o,$o),fconj,(P)),(Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: $o,Q: $o] :
      ( ~ aa($o,$o,aa($o,fun($o,$o),fdisj,(P)),(Q))
      | (P)
      | (Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: $o,P: $o] :
      ( ~ (Q)
      | aa($o,$o,aa($o,fun($o,$o),fdisj,(P)),(Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: $o,Q: $o] :
      ( ~ (P)
      | aa($o,$o,aa($o,fun($o,$o),fdisj,(P)),(Q)) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X8: A,Y: A] :
      ( ( X8 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X8),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X8: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X8),Y)
      | ( X8 = Y ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

tff(tfree_1,hypothesis,
    linorder(b) ).

% Conjectures (1)
tff(conj_0,conjecture,
    hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),v),aa(nat,nat,suc,zero_zero(nat)),vg,vh)),tia),vEBT_VEBT_vebt_succi(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),tia,vi),aTP_Lamp_a(option(nat),assn)) ).